mathdatech/python_reference
1
0
Fork 0
mirror of https://github.com/rasbt/python_reference.git synced 2024-12-18 00:00:19 +00:00
Code Issues Packages Projects Releases Wiki Activity
59851864c4
python_reference/tutorials/matrix_cheatsheet_only.html
rasbt 018a0b281e note about numpy in-place ops
2014-06-22 12:39:16 -04:00

1199 lines
105 KiB
HTML
Raw Blame History

This file contains ambiguous Unicode characters

This file contains Unicode characters that might be confused with other characters. If you think that this is intentional, you can safely ignore this warning. Use the Escape button to reveal them.

<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
<html>
<head>
<meta http-equiv="content-type" content="text/html; charset=utf-8">
<title></title>
<meta name="generator" content="LibreOffice 4.2.4.2 (MacOSX)">
<meta name="created" content="0;0">
<meta name="changed" content="20140622;123733939958000">
<style type="text/css">
<!--
td p { color: #000000; font-family: "Verdana"; font-size: 12pt }
p { color: #000000; font-family: "Verdana"; font-size: 12pt }
-->
</style>
</head>
<body lang="en-US" text="#000000" dir="ltr" style="background: transparent">
<table cellpadding="2" cellspacing="0" style="page-break-before: always">
<col width="383">
<col width="387">
<col width="447">
<col width="469">
<col width="511">
<col width="690">
<tr>
<td height="59" bgcolor="#0066cc" style="border: none; padding: 0in">
<p align="center"><font color="#ffffff"><font face="Arial"><font size="5" style="font-size: 18pt"><b>Task</b></font></font></font></p>
</td>
<td bgcolor="#0066cc" style="border: none; padding: 0in">
<p align="center"><font color="#ffffff"><font face="Andale Mono"><font size="5" style="font-size: 18pt"><b>MATLAB/Octave</b></font></font></font></p>
</td>
<td bgcolor="#0066cc" style="border: none; padding: 0in">
<p align="center"><font color="#ffffff"><font face="Andale Mono"><font size="5" style="font-size: 18pt"><b>Python
NumPy</b></font></font></font></p>
</td>
<td bgcolor="#0066cc" style="border: none; padding: 0in">
<p align="center"><font color="#ffffff"><font face="Andale Mono"><font size="5" style="font-size: 18pt"><b>R</b></font></font></font></p>
</td>
<td bgcolor="#0066cc" style="border: none; padding: 0in">
<p align="center"><font color="#ffffff"><font face="Andale Mono"><font size="5" style="font-size: 18pt"><b>Julia</b></font></font></font></p>
</td>
<td bgcolor="#0066cc" style="border: none; padding: 0in">
<p align="center"><font color="#ffffff"><font face="Arial"><font size="5" style="font-size: 18pt"><b>Task</b></font></font></font></p>
</td>
</tr>
<tr>
<td colspan="6" height="70" bgcolor="#ffffff" style="border: none; padding: 0in">
<p align="center"><font face="Arial"><font size="4" style="font-size: 14pt"><b>CREATING
MATRICES</b></font></font></p>
</td>
</tr>
<tr valign="top">
<td height="158" bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: none; border-right: 1px solid #000000; padding-top: 0in; padding-bottom: 0in; padding-left: 0in; padding-right: 0.02in">
<p align="left"><font face="Arial"><font size="2" style="font-size: 10pt"><b>Creating
Matrices&nbsp;<br>(here: 3x3 matrix)</b></font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: none; border-right: 1px dotted #000000; padding-top: 0in; padding-bottom: 0in; padding-left: 0in; padding-right: 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>
A = [1 2 3; 4 5 6; 7 8 9]<br>A =<br>&nbsp;&nbsp; 1 &nbsp; 2 &nbsp;
3<br>&nbsp;&nbsp; 4 &nbsp; 5 &nbsp; 6<br>&nbsp;&nbsp; 7 &nbsp; 8 &nbsp;
9</font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: 1px dotted #000000; padding: 0in 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
A = np.array([ [1,2,3], [4,5,6], [7,8,9] ])<br><br><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
A<br>array([[1, 2, 3],<br>&nbsp;&nbsp; &nbsp; &nbsp; [4, 5, 6],<br>&nbsp;&nbsp;
&nbsp; &nbsp; [7, 8, 9]])</font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: 1px dotted #000000; padding: 0in 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
A = matrix(c(1,2,3,4,5,6,7,8,9),nrow=3,byrow=T)<br><br>
#
equivalent to
<br># A = matrix(1:9,nrow=3,byrow=T)

<br><br><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
A<br>[,1] [,2] [,3]<br>[1,] 1 2 3<br>[2,] 4 5 6<br>[3,] 7 8 9</font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: none; padding-top: 0in; padding-bottom: 0in; padding-left: 0.02in; padding-right: 0in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>
A=[1 2 3; 4 5 6; 7 8 9]<br>3x3 Array{Int64,2}:<br>1 2 3<br>4 5 6<br>7
8 9</font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: 1px solid #000000; border-right: 1px solid #000000; padding: 0in 0.02in">
<p align="left"><font face="Arial"><font size="2" style="font-size: 10pt"><b>Creating
Matrices&nbsp;<br>(here: 3x3 matrix)</b></font></font></p>
</td>
</tr>
<tr valign="top">
<td height="128" bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: none; border-right: 1px solid #000000; padding-top: 0in; padding-bottom: 0in; padding-left: 0in; padding-right: 0.02in">
<p align="left"><font face="Arial"><font size="2" style="font-size: 10pt"><b>Creating
an 1D column vector</b></font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: none; border-right: 1px dotted #000000; padding-top: 0in; padding-bottom: 0in; padding-left: 0in; padding-right: 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>
a = [1; 2; 3]<br>a =<br>&nbsp;&nbsp; 1<br>&nbsp;&nbsp; 2<br>&nbsp;&nbsp;
3</font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: 1px dotted #000000; padding: 0in 0.02in">
<p align="left"><font color="#ff420e"><font face="Andale Mono"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font></font>
<font face="Andale Mono"><font size="1" style="font-size: 8pt">a</font></font><font color="#2fff12">
</font><font color="#000000"><font face="Andale Mono"><font size="1" style="font-size: 8pt">=
np.array([1,2,3]).reshape(1,3)</font></font></font></p>
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><br></font></font><font color="#ff420e"><font face="Andale Mono"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font></font>
<font color="#000000"><font face="Andale Mono"><font size="1" style="font-size: 8pt">b.shape<br>(1,
3)</font></font></font></p>
<p align="left"><br>
</p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: 1px dotted #000000; padding: 0in 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
a = matrix(c(1,2,3), nrow=3, byrow=T)<br><br><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
a<br>[,1]<br>[1,] 1<br>[2,] 2<br>[3,] 3</font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: none; padding-top: 0in; padding-bottom: 0in; padding-left: 0.02in; padding-right: 0in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>
a=[1; 2; 3]<br>3-element Array{Int64,1}: <br>1<br>2<br>3</font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: 1px solid #000000; border-right: 1px solid #000000; padding: 0in 0.02in">
<p align="left"><font face="Arial"><font size="2" style="font-size: 10pt"><b>Creating
an 1D column vector</b></font></font></p>
</td>
</tr>
<tr valign="top">
<td height="128" bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: none; border-right: 1px solid #000000; padding-top: 0in; padding-bottom: 0in; padding-left: 0in; padding-right: 0.02in">
<p align="left"><font face="Arial"><font size="2" style="font-size: 10pt"><b>Creating
an<br>1D row vector</b></font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: none; border-right: 1px dotted #000000; padding-top: 0in; padding-bottom: 0in; padding-left: 0in; padding-right: 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>
b = [1 2 3]<br>b =<br>&nbsp;&nbsp; 1 &nbsp; 2 &nbsp; 3</font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: 1px dotted #000000; padding: 0in 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
b = np.array([1,2,3])<br><br><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
b<br>array([1, 2, 3])</font></font></p>
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt">#
note that numpy doesn't have <br># explicit “row-vectors”, but
1-D <br># arrays</font></font></p>
<p style="margin-bottom: 0in"><font face="Liberation Serif, serif"><font size="3" style="font-size: 12pt"><font color="#ff420e"><font face="Andale Mono"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font></font><font color="#2fff12">
</font><font color="#000000"><font face="Andale Mono"><font size="1" style="font-size: 8pt">b.shape</font></font></font></font></font></p>
<p style="margin-bottom: 0in"><font face="Andale Mono"><font size="1" style="font-size: 8pt">(3,)</font></font></p>
<p align="left"><br>
</p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: 1px dotted #000000; padding: 0in 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
b = matrix(c(1,2,3), ncol=3)<br><br><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
b<br>[,1] [,2] [,3]<br>[1,] 1 2 3</font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: none; padding-top: 0in; padding-bottom: 0in; padding-left: 0.02in; padding-right: 0in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>
b=[1 2 3] <br>1x3 Array{Int64,2}:<br>1 2 3<br><br># note that this
is a 2D array. <br># vectors in Julia are columns</font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: 1px solid #000000; border-right: 1px solid #000000; padding: 0in 0.02in">
<p align="left"><font face="Arial"><font size="2" style="font-size: 10pt"><b>Creating
an<br>1D row vector</b></font></font></p>
</td>
</tr>
<tr valign="top">
<td height="128" bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: none; border-right: 1px solid #000000; padding-top: 0in; padding-bottom: 0in; padding-left: 0in; padding-right: 0.02in">
<p align="left"><font face="Arial"><font size="2" style="font-size: 10pt"><b>Creating
a <br>random m x n matrix</b></font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: none; border-right: 1px dotted #000000; padding-top: 0in; padding-bottom: 0in; padding-left: 0in; padding-right: 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>
rand(3,2)<br>ans =<br>&nbsp;&nbsp; 0.21977 &nbsp; 0.10220<br>&nbsp;&nbsp;
0.38959 &nbsp; 0.69911<br>&nbsp;&nbsp; 0.15624 &nbsp; 0.65637</font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: 1px dotted #000000; padding: 0in 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
np.random.rand(3,2)<br>array([[ 0.29347865,&nbsp; 0.17920462],<br>&nbsp;&nbsp;
&nbsp; &nbsp; [ 0.51615758,&nbsp; 0.64593471],<br>&nbsp;&nbsp; &nbsp;
&nbsp; [ 0.01067605,&nbsp; 0.09692771]])</font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: 1px dotted #000000; padding: 0in 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
matrix(runif(3*2), ncol=2)<br>[,1] [,2]<br>[1,] 0.5675127
0.7751204<br>[2,] 0.3439412 0.5261893<br>[3,] 0.2273177 0.223438</font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: none; padding-top: 0in; padding-bottom: 0in; padding-left: 0.02in; padding-right: 0in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>
rand(3,2)<br>3x2 Array{Float64,2}:<br>0.36882 0.267725<br>0.571856
0.601524<br>0.848084 0.858935</font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: 1px solid #000000; border-right: 1px solid #000000; padding: 0in 0.02in">
<p align="left"><font face="Arial"><font size="2" style="font-size: 10pt"><b>Creating
a <br>random m x n matrix</b></font></font></p>
</td>
</tr>
<tr valign="top">
<td height="128" bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: none; border-right: 1px solid #000000; padding-top: 0in; padding-bottom: 0in; padding-left: 0in; padding-right: 0.02in">
<p align="left"><font face="Arial"><font size="2" style="font-size: 10pt"><b>Creating
a<br>zero&nbsp;m x n matrix&nbsp;</b></font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: none; border-right: 1px dotted #000000; padding-top: 0in; padding-bottom: 0in; padding-left: 0in; padding-right: 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>
zeros(3,2)<br>ans =<br>&nbsp;&nbsp; 0 &nbsp; 0<br>&nbsp;&nbsp; 0 &nbsp;
0<br>&nbsp;&nbsp; 0 &nbsp; 0</font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: 1px dotted #000000; padding: 0in 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
np.zeros((3,2))<br>array([[ 0.,&nbsp; 0.],<br>&nbsp;&nbsp; &nbsp;
&nbsp; [ 0.,&nbsp; 0.],<br>&nbsp;&nbsp; &nbsp; &nbsp; [ 0.,&nbsp;
0.]])</font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: 1px dotted #000000; padding: 0in 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
mat.or.vec(3, 2)<br>[,1] [,2]<br>[1,] 0 0<br>[2,] 0 0<br>[3,] 0 0</font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: none; padding-top: 0in; padding-bottom: 0in; padding-left: 0.02in; padding-right: 0in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>
zeros(3,2)<br>3x2 Array{Float64,2}:<br>0.0 0.0<br>0.0 0.0<br>0.0
0.0</font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: 1px solid #000000; border-right: 1px solid #000000; padding: 0in 0.02in">
<p align="left"><font face="Arial"><font size="2" style="font-size: 10pt"><b>Creating
a<br>zero&nbsp;m x n matrix&nbsp;</b></font></font></p>
</td>
</tr>
<tr valign="top">
<td height="128" bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: none; border-right: 1px solid #000000; padding-top: 0in; padding-bottom: 0in; padding-left: 0in; padding-right: 0.02in">
<p align="left"><font face="Arial"><font size="2" style="font-size: 10pt"><b>Creating
an<br>m x n matrix of ones</b></font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: none; border-right: 1px dotted #000000; padding-top: 0in; padding-bottom: 0in; padding-left: 0in; padding-right: 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>
ones(3,2)<br>ans =<br>&nbsp;&nbsp; 1 &nbsp; 1<br>&nbsp;&nbsp; 1 &nbsp;
1<br>&nbsp;&nbsp; 1 &nbsp; 1</font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: 1px dotted #000000; padding: 0in 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
np.ones((3,2))<br>array([[ 1.,&nbsp; 1.],<br>&nbsp;&nbsp; &nbsp; &nbsp;
[ 1.,&nbsp; 1.],<br>&nbsp;&nbsp; &nbsp; &nbsp; [ 1.,&nbsp; 1.]])</font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: 1px dotted #000000; padding: 0in 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
mat.or.vec(3, 2) + 1<br>[,1] [,2]<br>[1,] 1 1<br>[2,] 1 1<br>[3,]
1 1</font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: none; padding-top: 0in; padding-bottom: 0in; padding-left: 0.02in; padding-right: 0in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>
ones(3,2)<br>3x2 Array{Float64,2}:<br>1.0 1.0<br>1.0 1.0<br>1.0
1.0</font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: 1px solid #000000; border-right: 1px solid #000000; padding: 0in 0.02in">
<p align="left"><font face="Arial"><font size="2" style="font-size: 10pt"><b>Creating
an<br>m x n matrix of ones</b></font></font></p>
</td>
</tr>
<tr valign="top">
<td height="128" bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: none; border-right: 1px solid #000000; padding-top: 0in; padding-bottom: 0in; padding-left: 0in; padding-right: 0.02in">
<p align="left"><font face="Arial"><font size="2" style="font-size: 10pt"><b>Creating
an<br>identity matrix</b></font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: none; border-right: 1px dotted #000000; padding-top: 0in; padding-bottom: 0in; padding-left: 0in; padding-right: 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>
eye(3)<br>ans =<br>Diagonal Matrix<br>&nbsp;&nbsp; 1 &nbsp; 0 &nbsp;
0<br>&nbsp;&nbsp; 0 &nbsp; 1 &nbsp; 0<br>&nbsp;&nbsp; 0 &nbsp; 0 &nbsp;
1</font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: 1px dotted #000000; padding: 0in 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
np.eye(3)<br>array([[ 1.,&nbsp; 0.,&nbsp; 0.],<br>&nbsp;&nbsp; &nbsp;
&nbsp; [ 0.,&nbsp; 1.,&nbsp; 0.],<br>&nbsp;&nbsp; &nbsp; &nbsp; [
0.,&nbsp; 0.,&nbsp; 1.]])</font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: 1px dotted #000000; padding: 0in 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
diag(3)<br>[,1] [,2] [,3]<br>[1,] 1 0 0<br>[2,] 0 1 0<br>[3,] 0 0
1</font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: none; padding-top: 0in; padding-bottom: 0in; padding-left: 0.02in; padding-right: 0in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>
eye(3)<br>3x3 Array{Float64,2}:<br>1.0 0.0 0.0<br>0.0 1.0 0.0<br>0.0
0.0 1.0</font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: 1px solid #000000; border-right: 1px solid #000000; padding: 0in 0.02in">
<p align="left"><font face="Arial"><font size="2" style="font-size: 10pt"><b>Creating
an<br>identity matrix</b></font></font></p>
</td>
</tr>
<tr valign="top">
<td height="217" bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: none; border-right: 1px solid #000000; padding-top: 0in; padding-bottom: 0in; padding-left: 0in; padding-right: 0.02in">
<p align="left"><font face="Arial"><font size="2" style="font-size: 10pt"><b>Creating
a<br>diagonal matrix</b></font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: none; border-right: 1px dotted #000000; padding-top: 0in; padding-bottom: 0in; padding-left: 0in; padding-right: 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>
a = [1 2 3]<br><br><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>
diag(a)<br>ans =<br>Diagonal Matrix<br>&nbsp;&nbsp; 1 &nbsp; 0 &nbsp;
0<br>&nbsp;&nbsp; 0 &nbsp; 2 &nbsp; 0<br>&nbsp;&nbsp; 0 &nbsp; 0 &nbsp;
3</font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: 1px dotted #000000; padding: 0in 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
a = np.array([1,2,3])<br><br><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
np.diag(a)<br>array([[1, 0, 0],<br>&nbsp;&nbsp; &nbsp; &nbsp; [0,
2, 0],<br>&nbsp;&nbsp; &nbsp; &nbsp; [0, 0, 3]])</font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: 1px dotted #000000; padding: 0in 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
diag(1:3)<br>[,1] [,2] [,3]<br>[1,] 1 0 0<br>[2,] 0 2 0<br>[3,] 0
0 3</font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: none; padding-top: 0in; padding-bottom: 0in; padding-left: 0.02in; padding-right: 0in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>
a=[1, 2, 3] <br><br># added commas because julia<br># vectors are
columnar<br><br><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>
diagm(a)<br>3x3 Array{Int64,2}:<br>1 0 0<br>0 2 0<br>0 0 3</font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: 1px solid #000000; border-right: 1px solid #000000; padding: 0in 0.02in">
<p align="left"><font face="Arial"><font size="2" style="font-size: 10pt"><b>Creating
a<br>diagonal matrix</b></font></font></p>
</td>
</tr>
<tr>
<td colspan="6" height="128" bgcolor="#ffffff" style="border: none; padding: 0in">
<p align="center"><font face="Arial"><font size="4" style="font-size: 14pt"><b>ACCESSING
MATRIX ELEMENTS</b></font></font></p>
</td>
</tr>
<tr valign="top">
<td height="159" bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: none; border-right: 1px solid #000000; padding-top: 0in; padding-bottom: 0in; padding-left: 0in; padding-right: 0.02in">
<p align="left"><font face="Arial"><font size="2" style="font-size: 10pt"><b>Getting
the dimension<br>of a matrix<br>(here: 2D, rows x cols)</b></font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: none; border-right: 1px dotted #000000; padding-top: 0in; padding-bottom: 0in; padding-left: 0in; padding-right: 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>
A = [1 2 3; 4 5 6]<br>A =<br>&nbsp; &nbsp;1 &nbsp; 2 &nbsp; 3<br>&nbsp;
&nbsp;4 &nbsp; 5 &nbsp; 6<br><br><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>
size(A)<br>ans =<br>&nbsp; &nbsp;2 &nbsp; 3</font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: 1px dotted #000000; padding: 0in 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
A = np.array([ [1,2,3], [4,5,6] ])<br><br><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
A<br>array([[1, 2, 3],<br>&nbsp; &nbsp; &nbsp; &nbsp;[4, 5,
6]])<br><br><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
A.shape<br>(2, 3)</font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: 1px dotted #000000; padding: 0in 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
A = matrix(1:6,nrow=2,byrow=T)<br><br><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
A<br>[,1] [,2] [,3]<br>[1,] 1 2 3<br>[2,] 4 5 6
</font></font></p>
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
dim(A)<br>[1] 2 3</font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: none; padding-top: 0in; padding-bottom: 0in; padding-left: 0.02in; padding-right: 0in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>
A=[1 2 3; 4 5 6]<br>2x3 Array{Int64,2}:<br>1 2 3<br>4 5 6<br><br><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>
size(A)<br>(2,3)</font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: 1px solid #000000; border-right: 1px solid #000000; padding: 0in 0.02in">
<p align="left"><font face="Arial"><font size="2" style="font-size: 10pt"><b>Getting
the dimension<br>of a matrix<br>(here: 2D, rows x cols)</b></font></font></p>
</td>
</tr>
<tr valign="top">
<td height="240" bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: none; border-right: 1px solid #000000; padding-top: 0in; padding-bottom: 0in; padding-left: 0in; padding-right: 0.02in">
<p align="left"><font face="Arial"><font size="2" style="font-size: 10pt"><b>Selecting
rows&nbsp;</b></font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: none; border-right: 1px dotted #000000; padding-top: 0in; padding-bottom: 0in; padding-left: 0in; padding-right: 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>
A = [1 2 3; 4 5 6; 7 8 9]<br><br>% 1st row<br><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>
A(1,:)<br>ans =<br>&nbsp;&nbsp; 1 &nbsp; 2 &nbsp; 3<br><br>% 1st 2
rows<br><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>
A(1:2,:)<br>ans =<br>&nbsp;&nbsp; 1 &nbsp; 2 &nbsp; 3<br>&nbsp;&nbsp;
4 &nbsp; 5 &nbsp; 6</font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: 1px dotted #000000; padding: 0in 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
A = np.array([ [1,2,3], [4,5,6], [7,8,9] ])<br><br># 1st row<br><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
A[0,:]<br>array([1, 2, 3])<br><br># 1st 2 rows<br><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
A[0:2,:]<br>array([[1, 2, 3], [4, 5, 6]])</font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: 1px dotted #000000; padding: 0in 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
A = matrix(1:9,nrow=3,byrow=T)

<br><br># 1st row<br>

<font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
A[1,]<br>[1] 1 2 3<br><br>

# 1st 2 rows

<br><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
A[1:2,]<br>[,1] [,2] [,3]<br>[1,] 1 2 3<br>[2,] 4 5 6</font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: none; padding-top: 0in; padding-bottom: 0in; padding-left: 0.02in; padding-right: 0in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>
A=[1 2 3; 4 5 6; 7 8 9]; <br>#semicolon suppresses output<br><br>#1st
row<br><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>
A[1,:] <br>1x3 Array{Int64,2}:<br>1 2 3<br><br>#1st 2 rows<br><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>
A[1:2,:] <br>2x3 Array{Int64,2}:<br>1 2 3<br>4 5 6</font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: 1px solid #000000; border-right: 1px solid #000000; padding: 0in 0.02in">
<p align="left"><font face="Arial"><font size="2" style="font-size: 10pt"><b>Selecting
rows&nbsp;</b></font></font></p>
</td>
</tr>
<tr valign="top">
<td height="304" bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: none; border-right: 1px solid #000000; padding-top: 0in; padding-bottom: 0in; padding-left: 0in; padding-right: 0.02in">
<p align="left"><font face="Arial"><font size="2" style="font-size: 10pt"><b>Selecting
columns</b></font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: none; border-right: 1px dotted #000000; padding-top: 0in; padding-bottom: 0in; padding-left: 0in; padding-right: 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>
A = [1 2 3; 4 5 6; 7 8 9]<br><br>% 1st column<br><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>
A(:,1)<br>ans =<br>&nbsp;&nbsp; 1<br>&nbsp;&nbsp; 4<br>&nbsp;&nbsp;
7<br><br>% 1st 2 columns<br><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>
A(:,1:2)<br>ans =<br>&nbsp;&nbsp; 1 &nbsp; 2<br>&nbsp;&nbsp; 4 &nbsp;
5<br>&nbsp;&nbsp; 7 &nbsp; 8</font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: 1px dotted #000000; padding: 0in 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
A = np.array([ [1,2,3], [4,5,6], [7,8,9] ])<br><br># 1st column
(as row vector)<br><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
A[:,0]<br>array([1, 4, 7])<br><br># 1st column (as column
vector)<br><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
A[:,[0]]<br>array([[1],<br>&nbsp;&nbsp; &nbsp; &nbsp; [4],<br>&nbsp;&nbsp;
&nbsp; &nbsp; [7]])<br><br># 1st 2 columns<br><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
A[:,0:2]<br>array([[1, 2],&nbsp;<br>&nbsp; &nbsp; &nbsp; &nbsp;[4,
5],&nbsp;<br>&nbsp; &nbsp; &nbsp; &nbsp;[7, 8]])</font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: 1px dotted #000000; padding: 0in 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
A = matrix(1:9,nrow=3,byrow=T)


<br><br># 1st column as row
vector
<br><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
t(A[,1])<br>[,1] [,2] [,3]<br>[1,] 1 4 7

<br><br># 1st column
as column vector
<br><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
A[,1]<br>[1] 1 4 7

<br><br># 1st 2 columns
<br><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
A[,1:2]<br>[,1] [,2]<br>[1,] 1 2<br>[2,] 4 5<br>[3,] 7 8</font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: none; padding-top: 0in; padding-bottom: 0in; padding-left: 0.02in; padding-right: 0in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>
A=[1 2 3; 4 5 6; 7 8 9];<br><br>#1st column<br><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>
A[:,1] <br>3-element Array{Int64,1}:<br>1<br>4<br>7<br><br>#1st 2
columns<br><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>
A[:,1:2] <br>3x2 Array{Int64,2}:<br>1 2<br>4 5<br>7 8</font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: 1px solid #000000; border-right: 1px solid #000000; padding: 0in 0.02in">
<p align="left"><font face="Arial"><font size="2" style="font-size: 10pt"><b>Selecting
columns</b></font></font></p>
</td>
</tr>
<tr valign="top">
<td height="260" bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: none; border-right: 1px solid #000000; padding-top: 0in; padding-bottom: 0in; padding-left: 0in; padding-right: 0.02in">
<p align="left"><font face="Arial"><font size="2" style="font-size: 10pt"><b>Extracting
rows and columns by criteria<br><br>(here: get rows&nbsp;that have
value 9 in column 3)</b></font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: none; border-right: 1px dotted #000000; padding-top: 0in; padding-bottom: 0in; padding-left: 0in; padding-right: 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>
A = [1 2 3; 4 5 9; 7 8 9]<br>A =<br>&nbsp;&nbsp; 1 &nbsp; 2 &nbsp;
3<br>&nbsp;&nbsp; 4 &nbsp; 5 &nbsp; 9<br>&nbsp;&nbsp; 7 &nbsp; 8 &nbsp;
9<br><br><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>
A(A(:,3) == 9,:)<br>ans =<br>&nbsp;&nbsp; 4 &nbsp; 5 &nbsp; 9<br>&nbsp;&nbsp;
7 &nbsp; 8 &nbsp; 9</font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: 1px dotted #000000; padding: 0in 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
A = np.array([ [1,2,3], [4,5,9], [7,8,9]])<br><br><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
A<br>array([[1, 2, 3],<br>&nbsp;&nbsp; &nbsp; &nbsp; [4, 5, 9],<br>&nbsp;&nbsp;
&nbsp; &nbsp; [7, 8, 9]])<br><br><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
A[A[:,2] == 9]<br>array([[4, 5, 9],<br>&nbsp;&nbsp; &nbsp; &nbsp;
[7, 8, 9]])</font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: 1px dotted #000000; padding: 0in 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
A = matrix(1:9,nrow=3,byrow=T)

<br><br><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
A<br>[,1] [,2] [,3]<br>[1,] 1 2 3<br>[2,] 4 5 9<br>[3,] 7 8
9

<br><br><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
matrix(A[A[,3]==9], ncol=3)<br>[,1] [,2] [,3]<br>[1,] 4 5 9<br>[2,]
7 8 9</font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: none; padding-top: 0in; padding-bottom: 0in; padding-left: 0.02in; padding-right: 0in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>
A=[1 2 3; 4 5 9; 7 8 9]<br>3x3 Array{Int64,2}:<br>1 2 3<br>4 5 9<br>7
8 9<br><br># use '.==' for<br># element-wise check<br><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>
A[ A[:,3] .==9, :] <br>2x3 Array{Int64,2}:<br>4 5 9<br>7 8 9</font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: 1px solid #000000; border-right: 1px solid #000000; padding: 0in 0.02in">
<p align="left"><font face="Arial"><font size="2" style="font-size: 10pt"><b>Extracting
rows and columns by criteria<br><br>(here: get rows&nbsp;that have
value 9 in column 3)</b></font></font></p>
</td>
</tr>
<tr valign="top">
<td height="118" bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: none; border-right: 1px solid #000000; padding-top: 0in; padding-bottom: 0in; padding-left: 0in; padding-right: 0.02in">
<p align="left"><font face="Arial"><font size="2" style="font-size: 10pt"><b>Accessing
elements<br>(here: 1st element)</b></font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: none; border-right: 1px dotted #000000; padding-top: 0in; padding-bottom: 0in; padding-left: 0in; padding-right: 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>
A = [1 2 3; 4 5 6; 7 8 9]<br><br><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>
A(1,1)<br>ans =&nbsp; 1</font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: 1px dotted #000000; padding: 0in 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
A = np.array([ [1,2,3], [4,5,6], [7,8,9] ])<br><br><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
A[0,0]<br>1</font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: 1px dotted #000000; padding: 0in 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
A = matrix(c(1,2,3,4,5,9,7,8,9),nrow=3,byrow=T)
<br><br>
<font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
A[1,1]<br>[1] 1</font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: none; padding-top: 0in; padding-bottom: 0in; padding-left: 0.02in; padding-right: 0in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>
A=[1 2 3; 4 5 6; 7 8 9];<br><br><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>
A[1,1]<br>1</font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: 1px solid #000000; border-right: 1px solid #000000; padding: 0in 0.02in">
<p align="left"><font face="Arial"><font size="2" style="font-size: 10pt"><b>Accessing
elements<br>(here: 1st element)</b></font></font></p>
</td>
</tr>
<tr>
<td colspan="6" height="128" bgcolor="#ffffff" style="border: none; padding: 0in">
<p align="center"><font face="Arial"><font size="4" style="font-size: 14pt"><b>MANIPULATING
SHAPE AND DIMENSIONS</b></font></font></p>
</td>
</tr>
<tr valign="top">
<td height="173" bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: none; border-right: 1px solid #000000; padding-top: 0in; padding-bottom: 0in; padding-left: 0in; padding-right: 0.02in">
<p align="left"><font face="Arial"><font size="2" style="font-size: 10pt"><b>Converting&nbsp;<br>row
to column vectors</b></font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: none; border-right: 1px dotted #000000; padding-top: 0in; padding-bottom: 0in; padding-left: 0in; padding-right: 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>
b = [1 2 3]
<br><br><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>
b = b'<br>b =<br>&nbsp;&nbsp; 1<br>&nbsp;&nbsp; 2<br>&nbsp;&nbsp;
3</font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: 1px dotted #000000; padding: 0in 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
b = np.array([1, 2, 3])<br><br><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
b = b[np.newaxis].T<br># alternatively <br>#&nbsp;b =
b[:,np.newaxis]<br><br><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
b<br>array([[1],<br>&nbsp;&nbsp; &nbsp; &nbsp; [2],<br>&nbsp;&nbsp;
&nbsp; &nbsp; [3]])</font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: 1px dotted #000000; padding: 0in 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
b = matrix(c(1,2,3), ncol=3)<br><br><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
t(b)<br>[,1]<br>[1,] 1<br>[2,] 2<br>[3,] 3</font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: none; padding-top: 0in; padding-bottom: 0in; padding-left: 0.02in; padding-right: 0in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>
b=vec([1 2 3])<br>3-element Array{Int64,1}:<br>1<br>2<br>3</font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: 1px solid #000000; border-right: 1px solid #000000; padding: 0in 0.02in">
<p align="left"><font face="Arial"><font size="2" style="font-size: 10pt"><b>Converting&nbsp;<br>row
to column vectors</b></font></font></p>
</td>
</tr>
<tr valign="top">
<td height="275" bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: none; border-right: 1px solid #000000; padding-top: 0in; padding-bottom: 0in; padding-left: 0in; padding-right: 0.02in">
<p align="left"><font face="Arial"><font size="2" style="font-size: 10pt"><b>Reshaping
Matrices<br><br>(here: 3x3 matrix to row vector)</b></font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: none; border-right: 1px dotted #000000; padding-top: 0in; padding-bottom: 0in; padding-left: 0in; padding-right: 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>
A = [1 2 3; 4 5 6; 7 8 9]<br>A =<br>&nbsp;&nbsp; 1 &nbsp; 2 &nbsp;
3<br>&nbsp;&nbsp; 4 &nbsp; 5 &nbsp; 6<br>&nbsp;&nbsp; 7 &nbsp; 8 &nbsp;
9<br><br><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>
total_elements = numel(A)<br><br><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>
B = reshape(A,1,total_elements)&nbsp;<br>% or reshape(A,1,9)<br>B
=<br>&nbsp;&nbsp; 1 &nbsp; 4 &nbsp; 7 &nbsp; 2 &nbsp; 5 &nbsp; 8 &nbsp;
3 &nbsp; 6 &nbsp; 9</font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: 1px dotted #000000; padding: 0in 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
A = np.array([[1,2,3],[4,5,6],[7,8,9]])<br><br><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
A<br>array([[1, 2, 3],<br>&nbsp;&nbsp; &nbsp; &nbsp; [4, 5, 9],<br>&nbsp;&nbsp;
&nbsp; &nbsp; [7, 8, 9]])<br><br><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
total_elements = np.prod(A.shape)</font></font></p>
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
B = A.reshape(1, total_elements)&nbsp;<br><br># alternative
shortcut:<br># A.reshape(1,-1)<br><br><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
B<br>array([[1, 2, 3, 4, 5, 6, 7, 8, 9]])</font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: 1px dotted #000000; padding: 0in 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
A = matrix(1:9,nrow=3,byrow=T)

<br><br><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
A<br>[,1] [,2] [,3]<br>[1,] 1 2 3<br>[2,] 4 5 6<br>[3,] 7 8 9
<br><br><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
total_elements = dim(A)[1] * dim(A)[2]<br><br><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
B = matrix(A, ncol=total_elements)<br><br><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
B<br>[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9]<br>[1,] 1 4 7 2
5 8 3 6 9</font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: none; padding-top: 0in; padding-bottom: 0in; padding-left: 0.02in; padding-right: 0in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>
A=[1 2 3; 4 5 6; 7 8 9]<br>3x3 Array{Int64,2}:<br>1 2 3<br>4 5 6<br>7
8 9<br><br><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>
total_elements=length(A)<br>9<br><br><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>B=reshape(A,1,total_elements)<br>1x9
Array{Int64,2}:<br>1 4 7 2 5 8 3 6 9</font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: 1px solid #000000; border-right: 1px solid #000000; padding: 0in 0.02in">
<p align="left"><font face="Arial"><font size="2" style="font-size: 10pt"><b>Reshaping
Matrices<br><br>(here: 3x3 matrix to row vector)</b></font></font></p>
</td>
</tr>
<tr valign="top">
<td height="240" bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: none; border-right: 1px solid #000000; padding-top: 0in; padding-bottom: 0in; padding-left: 0in; padding-right: 0.02in">
<p align="left"><font face="Arial"><font size="2" style="font-size: 10pt"><b>Concatenating
matrices</b></font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: none; border-right: 1px dotted #000000; padding-top: 0in; padding-bottom: 0in; padding-left: 0in; padding-right: 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>
A = [1 2 3; 4 5 6]<br><br><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>
B = [7 8 9; 10 11 12]<br><br><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>
C = [A; B]<br>&nbsp; &nbsp;&nbsp;1&nbsp; &nbsp; 2&nbsp; &nbsp; 3<br>&nbsp;
&nbsp; 4&nbsp; &nbsp; 5&nbsp; &nbsp; 6<br>&nbsp; &nbsp; 7&nbsp; &nbsp;
8&nbsp; &nbsp; 9<br>&nbsp;&nbsp; 10 &nbsp; 11 &nbsp; 12</font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: 1px dotted #000000; padding: 0in 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
A = np.array([[1, 2, 3], [4, 5, 6]])<br><br><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
B = np.array([[7, 8, 9],[10,11,12]])<br><br><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
C = np.concatenate((A, B), axis=0)<br><br><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
C<br>array([[ 1, 2, 3],&nbsp;<br>&nbsp; &nbsp; &nbsp; &nbsp;[ 4,
5, 6],&nbsp;<br>&nbsp; &nbsp; &nbsp; &nbsp;[ 7, 8, 9],&nbsp;<br>&nbsp;
&nbsp; &nbsp; &nbsp;[10, 11, 12]])</font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: 1px dotted #000000; padding: 0in 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
A = matrix(1:6,nrow=2,byrow=T)<br><br><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
B = matrix(7:12,nrow=2,byrow=T)<br><br><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
C = rbind(A,B)<br><br><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
C<br>[,1] [,2] [,3]<br>[1,] 1 2 3<br>[2,] 4 5 6<br>[3,] 7 8 9<br>[4,]
10 11 12</font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: none; padding-top: 0in; padding-bottom: 0in; padding-left: 0.02in; padding-right: 0in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>
A=[1 2 3; 4 5 6];<br><br><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>
B=[7 8 9; 10 11 12];<br><br><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>
C=[A; B]<br>4x3 Array{Int64,2}:<br>1 2 3<br>4 5 6<br>7 8 9<br>10
11 12</font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: 1px solid #000000; border-right: 1px solid #000000; padding: 0in 0.02in">
<p align="left"><font face="Arial"><font size="2" style="font-size: 10pt"><b>Concatenating
matrices</b></font></font></p>
</td>
</tr>
<tr valign="top">
<td height="256" bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: none; border-right: 1px solid #000000; padding-top: 0in; padding-bottom: 0in; padding-left: 0in; padding-right: 0.02in">
<p align="left"><font face="Arial"><font size="2" style="font-size: 10pt"><b>Stacking&nbsp;<br>vectors
and matrices</b></font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: none; border-right: 1px dotted #000000; padding-top: 0in; padding-bottom: 0in; padding-left: 0in; padding-right: 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>
a = [1 2 3]<br><br><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>
b = [4 5 6]<br><br><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>
c = [a' b']<br>c =<br>&nbsp;&nbsp; 1 &nbsp; 4<br>&nbsp;&nbsp; 2 &nbsp;
5<br>&nbsp;&nbsp; 3 &nbsp; 6<br><br><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>
c = [a; b]<br>c =<br>&nbsp;&nbsp; 1 &nbsp; 2 &nbsp; 3<br>&nbsp;&nbsp;
4 &nbsp; 5 &nbsp; 6</font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: 1px dotted #000000; padding: 0in 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
a = np.array([1,2,3])<br><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
b = np.array([4,5,6])<br><br><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
np.c_[a,b]<br>array([[1, 4],<br>&nbsp;&nbsp; &nbsp; &nbsp; [2,
5],<br>&nbsp;&nbsp; &nbsp; &nbsp; [3, 6]])<br><br><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
np.r_[a,b]<br>array([[1, 2, 3],<br>&nbsp;&nbsp; &nbsp; &nbsp; [4,
5, 6]])</font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: 1px dotted #000000; padding: 0in 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
a = matrix(1:3, ncol=3)<br><br><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
b = matrix(4:6, ncol=3)<br><br><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
matrix(rbind(A, B), ncol=2)<br>[,1] [,2]<br>[1,] 1 5<br>[2,] 4
3
<br><br>
<font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
rbind(A,B)<br>[,1] [,2] [,3]<br>[1,] 1 2 3<br>[2,] 4 5 6</font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: none; padding-top: 0in; padding-bottom: 0in; padding-left: 0.02in; padding-right: 0in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>
a=[1 2 3];<br><br><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>
b=[4 5 6];<br><br><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>
c=[a' b']<br>3x2 Array{Int64,2}:<br>1 4<br>2 5<br>3 6<br><br><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>
c=[a; b]<br>2x3 Array{Int64,2}:<br>1 2 3<br>4 5 6</font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: 1px solid #000000; border-right: 1px solid #000000; padding: 0in 0.02in">
<p align="left"><font face="Arial"><font size="2" style="font-size: 10pt"><b>Stacking&nbsp;<br>vectors
and matrices</b></font></font></p>
</td>
</tr>
<tr>
<td colspan="6" height="114" bgcolor="#ffffff" style="border: none; padding: 0in">
<p align="center"><font face="Arial"><font size="4" style="font-size: 14pt"><b>BASIC
MATRIX OPERATIONS</b></font></font></p>
</td>
</tr>
<tr valign="top">
<td height="253" bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: none; border-right: 1px solid #000000; padding-top: 0in; padding-bottom: 0in; padding-left: 0in; padding-right: 0.02in">
<p align="left"><font face="Arial"><font size="2" style="font-size: 10pt"><b>Matrix-scalar<br>operations</b></font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: none; border-right: 1px dotted #000000; padding-top: 0in; padding-bottom: 0in; padding-left: 0in; padding-right: 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>&nbsp;A
= [1 2 3; 4 5 6; 7 8 9]<br><br><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>
A * 2<br>ans =<br>&nbsp; &nbsp; 2&nbsp; &nbsp; 4&nbsp; &nbsp; 6<br>&nbsp;
&nbsp; 8 &nbsp; 10 &nbsp; 12<br>&nbsp;&nbsp; 14 &nbsp; 16 &nbsp;
18<br><br><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>
A + 2<br><br><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>
A - 2<br><br><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>
A / 2</font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: 1px dotted #000000; padding: 0in 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
A = np.array([ [1,2,3], [4,5,6], [7,8,9] ])<br><br><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
A * 2<br>array([[ 2,&nbsp; 4,&nbsp; 6],<br>&nbsp;&nbsp; &nbsp; &nbsp;
[ 8, 10, 12],<br>&nbsp;&nbsp; &nbsp; &nbsp; [14, 16, 18]])<br><br><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
A + 2<br><br><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
A - 2<br><br><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
A / 2</font></font></p>
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt">#
Note that NumPy was optimized for<br># in-place assignments<br>#
e.g., A += A instead of <br># A = A + A</font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: 1px dotted #000000; padding: 0in 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
A = matrix(1:9, nrow=3, byrow=T)<br><br><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
A * 2<br>[,1] [,2] [,3]<br>[1,] 2 4 6<br>[2,] 8 10 12<br>[3,] 14
16 18
</font></font></p>
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
A + 2<br><br><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
A - 2<br><br><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
A / 2</font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: none; padding-top: 0in; padding-bottom: 0in; padding-left: 0.02in; padding-right: 0in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>
A=[1 2 3; 4 5 6; 7 8 9];<br><br># elementwise operator<br><br><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>
A .* 2 <br>3x3 Array{Int64,2}:<br>2 4 6<br>8 10 12<br>14 16 18
<br><br><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>
A .+ 2;<br><br><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>
A .- 2;<br><br><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>
A ./ 2;</font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: 1px solid #000000; border-right: 1px solid #000000; padding: 0in 0.02in">
<p align="left"><font face="Arial"><font size="2" style="font-size: 10pt"><b>Matrix-scalar<br>operations</b></font></font></p>
</td>
</tr>
<tr valign="top">
<td height="165" bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: none; border-right: 1px solid #000000; padding-top: 0in; padding-bottom: 0in; padding-left: 0in; padding-right: 0.02in">
<p align="left"><font face="Arial"><font size="2" style="font-size: 10pt"><b>Matrix-matrix<br>multiplication</b></font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: none; border-right: 1px dotted #000000; padding-top: 0in; padding-bottom: 0in; padding-left: 0in; padding-right: 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>&nbsp;A
= [1 2 3; 4 5 6; 7 8 9]<br><br><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>
A * A<br>ans =<br>&nbsp; &nbsp; 30&nbsp; &nbsp; 36&nbsp; &nbsp;
42<br>&nbsp; &nbsp; 66&nbsp; &nbsp; 81&nbsp; &nbsp; 96<br>&nbsp;&nbsp;
102 &nbsp; 126 &nbsp; 150</font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: 1px dotted #000000; padding: 0in 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
A = np.array([ [1,2,3], [4,5,6], [7,8,9] ])<br><br><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
np.dot(A,A) # or A.dot(A)<br>array([[ 30,&nbsp; 36,&nbsp; 42],<br>&nbsp;&nbsp;
&nbsp; &nbsp; [ 66,&nbsp; 81,&nbsp; 96],<br>&nbsp;&nbsp; &nbsp; &nbsp;
[102, 126, 150]])</font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: 1px dotted #000000; padding: 0in 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
A = matrix(1:9, nrow=3, byrow=T)<br><br><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
A %*% A<br>[,1] [,2] [,3]<br>[1,] 30 36 42<br>[2,] 66 81 96<br>[3,]
102 126 150</font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: none; padding-top: 0in; padding-bottom: 0in; padding-left: 0.02in; padding-right: 0in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>
A=[1 2 3; 4 5 6; 7 8 9];<br><br><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>
A * A<br>3x3 Array{Int64,2}:<br>30 36 42<br>66 81 96<br>102 126
150</font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: 1px solid #000000; border-right: 1px solid #000000; padding: 0in 0.02in">
<p align="left"><font face="Arial"><font size="2" style="font-size: 10pt"><b>Matrix-matrix<br>multiplication</b></font></font></p>
</td>
</tr>
<tr valign="top">
<td height="190" bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: none; border-right: 1px solid #000000; padding-top: 0in; padding-bottom: 0in; padding-left: 0in; padding-right: 0.02in">
<p align="left"><font face="Arial"><font size="2" style="font-size: 10pt"><b>Matrix-vector<br>multiplication</b></font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: none; border-right: 1px dotted #000000; padding-top: 0in; padding-bottom: 0in; padding-left: 0in; padding-right: 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>
A = [1 2 3; 4 5 6; 7 8 9]<br><br><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>
b = [ 1; 2; 3 ]<br><br><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>
A * b<br>ans =<br>&nbsp;&nbsp; 14<br>&nbsp;&nbsp; 32<br>&nbsp;&nbsp;
50</font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: 1px dotted #000000; padding: 0in 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
A = np.array([ [1,2,3], [4,5,6], [7,8,9] ])<br><br><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
b = np.array([ [1], [2], [3] ])<br><br><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
np.dot(A,b) # or A.dot(b)<br><br>array([[14], [32], [50]])</font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: 1px dotted #000000; padding: 0in 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
A = matrix(1:9, ncol=3)<br><br><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
b = matrix(1:3, nrow=3)<br><br>

<font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
t(b %*% A)<br>[,1]<br>[1,] 14<br>[2,] 32<br>[3,] 50</font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: none; padding-top: 0in; padding-bottom: 0in; padding-left: 0.02in; padding-right: 0in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>
A=[1 2 3; 4 5 6; 7 8 9];<br><br><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>
b=[1; 2; 3];<br><br><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>
A*b<br>3-element Array{Int64,1}:<br>14<br>32<br>50</font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: 1px solid #000000; border-right: 1px solid #000000; padding: 0in 0.02in">
<p align="left"><font face="Arial"><font size="2" style="font-size: 10pt"><b>Matrix-vector<br>multiplication</b></font></font></p>
</td>
</tr>
<tr valign="top">
<td height="240" bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: none; border-right: 1px solid #000000; padding-top: 0in; padding-bottom: 0in; padding-left: 0in; padding-right: 0.02in">
<p align="left"><font face="Arial"><font size="2" style="font-size: 10pt"><b>Element-wise&nbsp;<br>matrix-matrix&nbsp;operations</b></font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: none; border-right: 1px dotted #000000; padding-top: 0in; padding-bottom: 0in; padding-left: 0in; padding-right: 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>&nbsp;A
= [1 2 3; 4 5 6; 7 8 9]<br><br><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>
A .* A<br>ans =<br>&nbsp; &nbsp; 1&nbsp; &nbsp; 4&nbsp; &nbsp;
9<br>&nbsp;&nbsp; 16 &nbsp; 25 &nbsp; 36<br>&nbsp;&nbsp; 49 &nbsp;
64 &nbsp; 81<br><br><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>
A .+ A<br><br><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>
A .- A<br><br><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>
A ./ A</font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: 1px dotted #000000; padding: 0in 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
A = np.array([ [1,2,3], [4,5,6], [7,8,9] ])<br><br><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
A * A<br>array([[ 1,&nbsp; 4,&nbsp; 9],<br>&nbsp;&nbsp; &nbsp; &nbsp;
[16, 25, 36],<br>&nbsp;&nbsp; &nbsp; &nbsp; [49, 64, 81]])<br><br><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
A + A<br><br><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
A - A<br><br><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
A / A</font></font></p>
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt">#
Note that NumPy was optimized for<br># in-place assignments<br>#
e.g., A += A instead of <br># A = A + A</font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: 1px dotted #000000; padding: 0in 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
A = matrix(1:9, nrow=3, byrow=T)
<br><br><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
A * A<br>[,1] [,2] [,3]<br>[1,] 1 4 9<br>[2,] 16 25 36<br>[3,] 49
64 81

<br><br><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
A + A<br><br><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
A - A<br><br><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
A / A</font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: none; padding-top: 0in; padding-bottom: 0in; padding-left: 0.02in; padding-right: 0in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>
A=[1 2 3; 4 5 6; 7 8 9];<br><br><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>
A .* A<br>3x3 Array{Int64,2}:<br>1 4 9<br>16 25 36<br>49 64 81
<br><br><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>
A .+ A; <br><br><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>
A .- A; <br><br><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>
A ./ A;</font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: 1px solid #000000; border-right: 1px solid #000000; padding: 0in 0.02in">
<p align="left"><font face="Arial"><font size="2" style="font-size: 10pt"><b>Element-wise&nbsp;<br>matrix-matrix&nbsp;operations</b></font></font></p>
</td>
</tr>
<tr valign="top">
<td height="164" bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: none; border-right: 1px solid #000000; padding-top: 0in; padding-bottom: 0in; padding-left: 0in; padding-right: 0.02in">
<p align="left"><font face="Arial"><font size="2" style="font-size: 10pt"><b>Matrix
elements&nbsp;to power n <br><br>(here: individual elements
squared)</b></font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: none; border-right: 1px dotted #000000; padding-top: 0in; padding-bottom: 0in; padding-left: 0in; padding-right: 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>&nbsp;A
= [1 2 3; 4 5 6; 7 8 9]<br><br><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>
A.^2<br>ans =<br>&nbsp; &nbsp; 1&nbsp; &nbsp; 4&nbsp; &nbsp; 9<br>&nbsp;&nbsp;
16 &nbsp; 25 &nbsp; 36<br>&nbsp;&nbsp; 49 &nbsp; 64 &nbsp; 81</font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: 1px dotted #000000; padding: 0in 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
A = np.array([ [1,2,3], [4,5,6], [7,8,9] ])<br><br><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
np.power(A,2)<br>array([[ 1,&nbsp; 4,&nbsp; 9],<br>&nbsp;&nbsp; &nbsp;
&nbsp; [16, 25, 36],<br>&nbsp;&nbsp; &nbsp; &nbsp; [49, 64, 81]])</font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: 1px dotted #000000; padding: 0in 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
A = matrix(1:9, nrow=3, byrow=T)<br><br><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
A ^ 2<br>[,1] [,2] [,3]<br>[1,] 1 4 9<br>[2,] 16 25 36<br>[3,] 49
64 81</font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: none; padding-top: 0in; padding-bottom: 0in; padding-left: 0.02in; padding-right: 0in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>
A=[1 2 3; 4 5 6; 7 8 9];<br><br><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>
A .^ 2<br>3x3 Array{Int64,2}:<br>1 4 9<br>16 25 36<br>49 64 81</font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: 1px solid #000000; border-right: 1px solid #000000; padding: 0in 0.02in">
<p align="left"><font face="Arial"><font size="2" style="font-size: 10pt"><b>Matrix
elements&nbsp;to power n <br><br>(here: individual elements
squared)</b></font></font></p>
</td>
</tr>
<tr valign="top">
<td height="225" bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: none; border-right: 1px solid #000000; padding-top: 0in; padding-bottom: 0in; padding-left: 0in; padding-right: 0.02in">
<p align="left"><font face="Arial"><font size="2" style="font-size: 10pt"><b>Matrix
to power n<br><br>(here: matrix-matrix&nbsp;multiplication with
itself)</b></font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: none; border-right: 1px dotted #000000; padding-top: 0in; padding-bottom: 0in; padding-left: 0in; padding-right: 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>&nbsp;A
= [1 2 3; 4 5 6; 7 8 9]<br><br><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>
A ^ 2<br>ans =<br>&nbsp; &nbsp; 30&nbsp; &nbsp; 36&nbsp; &nbsp;
42<br>&nbsp; &nbsp; 66&nbsp; &nbsp; 81&nbsp; &nbsp; 96<br>&nbsp;&nbsp;
102 &nbsp; 126 &nbsp; 150</font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: 1px dotted #000000; padding: 0in 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
A = np.array([ [1,2,3], [4,5,6], [7,8,9] ])<br><br><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
np.linalg.matrix_power(A,2)<br>array([[ 30,&nbsp; 36,&nbsp;
42],<br>&nbsp;&nbsp; &nbsp; &nbsp; [ 66,&nbsp; 81,&nbsp; 96],<br>&nbsp;&nbsp;
&nbsp; &nbsp; [102, 126, 150]])</font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: 1px dotted #000000; padding: 0in 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
A = matrix(1:9, ncol=3)<br><br>
# requires the ‘expm’
package
<br><br><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
install.packages('expm')
<br><br><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
library(expm)
<br><br>
<font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
A %^% 2<br>[,1] [,2] [,3]<br>[1,] 30 66 102<br>[2,] 36 81 126<br>[3,]
42 96 150</font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: none; padding-top: 0in; padding-bottom: 0in; padding-left: 0.02in; padding-right: 0in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>
A=[1 2 3; 4 5 6; 7 8 9];<br><br><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>
A ^ 2<br>3x3 Array{Int64,2}:<br>30 36 42<br>66 81 96<br>102 126
150</font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: 1px solid #000000; border-right: 1px solid #000000; padding: 0in 0.02in">
<p align="left"><font face="Arial"><font size="2" style="font-size: 10pt"><b>Matrix
to power n<br><br>(here: matrix-matrix&nbsp;multiplication with
itself)</b></font></font></p>
</td>
</tr>
<tr valign="top">
<td height="172" bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: none; border-right: 1px solid #000000; padding-top: 0in; padding-bottom: 0in; padding-left: 0in; padding-right: 0.02in">
<p align="left"><font face="Arial"><font size="2" style="font-size: 10pt"><b>Matrix
transpose</b></font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: none; border-right: 1px dotted #000000; padding-top: 0in; padding-bottom: 0in; padding-left: 0in; padding-right: 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>&nbsp;A
= [1 2 3; 4 5 6; 7 8 9]<br><br><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>
A'<br>ans =<br>&nbsp;&nbsp; 1 &nbsp; 4 &nbsp; 7<br>&nbsp;&nbsp; 2
&nbsp; 5 &nbsp; 8<br>&nbsp;&nbsp; 3 &nbsp; 6 &nbsp; 9</font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: 1px dotted #000000; padding: 0in 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
A = np.array([ [1,2,3], [4,5,6], [7,8,9] ])<br><br><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
A.T<br>array([[1, 4, 7],<br>&nbsp;&nbsp; &nbsp; &nbsp; [2, 5,
8],<br>&nbsp;&nbsp; &nbsp; &nbsp; [3, 6, 9]])</font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: 1px dotted #000000; padding: 0in 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
A = matrix(1:9, nrow=3, byrow=T)
<br><br><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
t(A)<br>[,1] [,2] [,3]<br>[1,] 1 4 7<br>[2,] 2 5 8<br>[3,] 3 6 9</font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: none; padding-top: 0in; padding-bottom: 0in; padding-left: 0.02in; padding-right: 0in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>
A=[1 2 3; 4 5 6; 7 8 9]<br>3x3 Array{Int64,2}:<br>1 2 3<br>4 5 6<br>7
8 9<br><br><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>
A'<br>3x3 Array{Int64,2}:<br>1 4 7<br>2 5 8<br>3 6 9</font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: 1px solid #000000; border-right: 1px solid #000000; padding: 0in 0.02in">
<p align="left"><font face="Arial"><font size="2" style="font-size: 10pt"><b>Matrix
transpose</b></font></font></p>
</td>
</tr>
<tr valign="top">
<td height="176" bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: none; border-right: 1px solid #000000; padding-top: 0in; padding-bottom: 0in; padding-left: 0in; padding-right: 0.02in">
<p align="left"><font face="Arial"><font size="2" style="font-size: 10pt"><b>Determinant
of a matrix:<br>&nbsp;A -&gt; |A|</b></font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: none; border-right: 1px dotted #000000; padding-top: 0in; padding-bottom: 0in; padding-left: 0in; padding-right: 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>
A = [6 1 1; 4 -2 5; 2 8 7]<br>A =<br>&nbsp;&nbsp; 6 &nbsp; 1 &nbsp;
1<br>&nbsp;&nbsp; 4&nbsp; -2 &nbsp; 5<br>&nbsp;&nbsp; 2 &nbsp; 8 &nbsp;
7<br><br><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>
det(A)<br>ans = -306</font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: 1px dotted #000000; padding: 0in 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>&nbsp;A
= np.array([[6,1,1],[4,-2,5],[2,8,7]])<br><br><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
A<br>array([[ 6,&nbsp; 1,&nbsp; 1],<br>&nbsp;&nbsp; &nbsp; &nbsp;
[ 4, -2,&nbsp; 5],<br>&nbsp;&nbsp; &nbsp; &nbsp; [ 2,&nbsp; 8,&nbsp;
7]])<br><br><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
np.linalg.det(A)<br>-306.0</font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: 1px dotted #000000; padding: 0in 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
A = matrix(c(6,1,1,4,-2,5,2,8,7), nrow=3, byrow=T)<br><br><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
A<br>[,1] [,2] [,3]<br>[1,] 6 1 1<br>[2,] 4 -2 5<br>[3,] 2 8 7<br><br><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
det(A)<br>[1] -306</font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: none; padding-top: 0in; padding-bottom: 0in; padding-left: 0.02in; padding-right: 0in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>
A=[6 1 1; 4 -2 5; 2 8 7]<br>3x3 Array{Int64,2}:<br>6 1 1<br>4 -2
5<br>2 8 7<br><br><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>
det(A)<br>-306.0</font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: 1px solid #000000; border-right: 1px solid #000000; padding: 0in 0.02in">
<p align="left"><font face="Arial"><font size="2" style="font-size: 10pt"><b>Determinant
of a matrix:<br>&nbsp;A -&gt; |A|</b></font></font></p>
</td>
</tr>
<tr valign="top">
<td height="202" bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: none; border-right: 1px solid #000000; padding-top: 0in; padding-bottom: 0in; padding-left: 0in; padding-right: 0.02in">
<p align="left"><font face="Arial"><font size="2" style="font-size: 10pt"><b>Inverse
of a matrix</b></font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: none; border-right: 1px dotted #000000; padding-top: 0in; padding-bottom: 0in; padding-left: 0in; padding-right: 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>
A = [4 7; 2 6]<br>A =<br>&nbsp;&nbsp; 4 &nbsp; 7<br>&nbsp;&nbsp; 2
&nbsp; 6<br><br><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>
A_inv = inv(A)<br>A_inv =<br>&nbsp; &nbsp;0.60000&nbsp; -0.70000<br>&nbsp;
-0.20000 &nbsp; 0.40000</font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: 1px dotted #000000; padding: 0in 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
A = np.array([[4, 7], [2, 6]])<br><br><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
A<br>array([[4, 7],&nbsp;<br>&nbsp; &nbsp; &nbsp; &nbsp;[2,
6]])<br><br><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
A_inverse = np.linalg.inv(A)<br><br><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
A_inverse<br>array([[ 0.6, -0.7],&nbsp;<br>&nbsp; &nbsp; &nbsp;
&nbsp;[-0.2, 0.4]])</font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: 1px dotted #000000; padding: 0in 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
A = matrix(c(4,7,2,6), nrow=2, byrow=T)<br><br><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
A<br>[,1] [,2]<br>[1,] 4 7<br>[2,] 2 6<br><br><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
solve(A)<br>[,1] [,2]<br>[1,] 0.6 -0.7<br>[2,] -0.2 0.4</font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: none; padding-top: 0in; padding-bottom: 0in; padding-left: 0.02in; padding-right: 0in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>
A=[4 7; 2 6]<br>2x2 Array{Int64,2}:<br>4 7<br>2 6<br><br><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>
A_inv=inv(A)<br>2x2 Array{Float64,2}:<br>0.6 -0.7<br>-0.2 0.4</font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: 1px solid #000000; border-right: 1px solid #000000; padding: 0in 0.02in">
<p align="left"><font face="Arial"><font size="2" style="font-size: 10pt"><b>Inverse
of a matrix</b></font></font></p>
</td>
</tr>
<tr>
<td colspan="6" height="202" bgcolor="#ffffff" style="border: none; padding: 0in">
<p align="center"><font face="Arial"><font size="4" style="font-size: 14pt"><b>ADVANCED
MATRIX OPERATIONS</b></font></font></p>
</td>
</tr>
<tr valign="top">
<td height="311" bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: none; border-right: 1px solid #000000; padding-top: 0in; padding-bottom: 0in; padding-left: 0in; padding-right: 0.02in">
<p align="left"><font face="Arial"><font size="2" style="font-size: 10pt"><b>Calculating
the covariance matrix&nbsp;<br>of 3 random variables <br><br>(here:
covariances of the means&nbsp;<br>of x1, x2, and x3)</b></font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: none; border-right: 1px dotted #000000; padding-top: 0in; padding-bottom: 0in; padding-left: 0in; padding-right: 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>
x1 = [4.0000 4.2000 3.9000 4.3000 4.1000]’<br><br><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>
x2 = [2.0000 2.1000 2.0000 2.1000 2.2000]'<br><br><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>
x3 = [0.60000 0.59000 0.58000 0.62000 0.63000]’<br><br><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>
cov( [x1,x2,x3] )<br>ans =<br>&nbsp;&nbsp; 2.5000e-02 &nbsp;
7.5000e-03 &nbsp; 1.7500e-03<br>&nbsp;&nbsp; 7.5000e-03 &nbsp;
7.0000e-03 &nbsp; 1.3500e-03<br>&nbsp;&nbsp; 1.7500e-03 &nbsp;
1.3500e-03 &nbsp; 4.3000e-04</font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: 1px dotted #000000; padding: 0in 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
x1 = np.array([ 4, 4.2, 3.9, 4.3, 4.1])<br><br><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
x2 = np.array([ 2, 2.1, 2, 2.1, 2.2])<br><br><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
x3 = np.array([ 0.6, 0.59, 0.58, 0.62, 0.63])<br><br><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
np.cov([x1, x2, x3])<br>Array([[ 0.025 &nbsp;, &nbsp;0.0075 ,
&nbsp;0.00175],<br>&nbsp; &nbsp; &nbsp; &nbsp;[ 0.0075 , &nbsp;0.007
&nbsp;, &nbsp;0.00135],<br>&nbsp; &nbsp; &nbsp; &nbsp;[ 0.00175,
&nbsp;0.00135, &nbsp;0.00043]])</font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: 1px dotted #000000; padding: 0in 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
x1 = matrix(c(4, 4.2, 3.9, 4.3, 4.1), ncol=5)<br><br><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
x2 = matrix(c(2, 2.1, 2, 2.1, 2.2), ncol=5)<br><br><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
x3 = matrix(c(0.6, 0.59, 0.58, 0.62, 0.63), ncol=5)

<br><br><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
cov(matrix(c(x1, x2, x3), ncol=3))<br>[,1] [,2] [,3]<br>[1,]
0.02500 0.00750 0.00175<br>[2,] 0.00750 0.00700 0.00135<br>[3,]
0.00175 0.00135 0.00043</font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: none; padding-top: 0in; padding-bottom: 0in; padding-left: 0.02in; padding-right: 0in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>
x1=[4.0 4.2 3.9 4.3 4.1]';<br><br><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>
x2=[2. 2.1 2. 2.1 2.2]';<br><br><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>
x3=[0.6 .59 .58 .62 .63]';<br><br><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>
cov([x1 x2 x3])<br>3x3 Array{Float64,2}:<br>0.025 0.0075
0.00175<br>0.0075 0.007 0.00135<br>0.00175 0.00135 0.00043</font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: 1px solid #000000; border-right: 1px solid #000000; padding: 0in 0.02in">
<p align="left"><font face="Arial"><font size="2" style="font-size: 10pt"><b>Calculating
the covariance matrix&nbsp;<br>of 3 random variables <br><br>(here:
covariances of the means&nbsp;<br>of x1, x2, and x3)</b></font></font></p>
</td>
</tr>
<tr valign="top">
<td height="309" bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: none; border-right: 1px solid #000000; padding-top: 0in; padding-bottom: 0in; padding-left: 0in; padding-right: 0.02in">
<p align="left"><font face="Arial"><font size="2" style="font-size: 10pt"><b>Calculating&nbsp;<br>eigenvectors
and&nbsp;eigenvalues</b></font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: none; border-right: 1px dotted #000000; padding-top: 0in; padding-bottom: 0in; padding-left: 0in; padding-right: 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>
A = [3 1; 1 3]<br>A =<br>&nbsp;&nbsp; 3 &nbsp; 1<br>&nbsp;&nbsp; 1
&nbsp; 3<br><br><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>
[eig_vec,eig_val] = eig(A)<br>eig_vec =<br>&nbsp; -0.70711 &nbsp;
0.70711<br>&nbsp;&nbsp; 0.70711 &nbsp; 0.70711<br>eig_val
=<br>Diagonal Matrix<br>&nbsp;&nbsp; 2 &nbsp; 0<br>&nbsp;&nbsp; 0
&nbsp; 4</font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: 1px dotted #000000; padding: 0in 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
A = np.array([[3, 1], [1, 3]])<br><br><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
A<br>array([[3, 1],<br>&nbsp;&nbsp; &nbsp; &nbsp; [1, 3]])<br><br><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
eig_val, eig_vec = np.linalg.eig(A)<br><br><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
eig_val<br>array([ 4.,&nbsp; 2.])<br><br><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
eig_vec<br>Array([[ 0.70710678, -0.70710678],<br>&nbsp;&nbsp; &nbsp;
&nbsp; [ 0.70710678,&nbsp; 0.70710678]])</font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: 1px dotted #000000; padding: 0in 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
A = matrix(c(3,1,1,3), ncol=2)<br><br><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
A<br>[,1] [,2]<br>[1,] 3 1<br>[2,] 1 3<br><br><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
eigen(A)<br>$values<br>[1] 4 2<br><br>$vectors<br>[,1] [,2]<br>[1,]
0.7071068 -0.7071068<br>[2,] 0.7071068 0.7071068</font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: none; padding-top: 0in; padding-bottom: 0in; padding-left: 0.02in; padding-right: 0in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>
A=[3 1; 1 3]<br>2x2 Array{Int64,2}:<br>3 1<br>1 3<br><br><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>
(eig_vec,eig_val)=eig(a)<br>([2.0,4.0],<br>2x2
Array{Float64,2}:<br>-0.707107 0.707107<br>0.707107 0.707107)</font></font></p>
</td>
<td bgcolor="#cfe7f5" style="border-top: none; border-bottom: none; border-left: 1px solid #000000; border-right: 1px solid #000000; padding: 0in 0.02in">
<p align="left"><font face="Arial"><font size="2" style="font-size: 10pt"><b>Calculating&nbsp;<br>eigenvectors
and&nbsp;eigenvalues</b></font></font></p>
</td>
</tr>
<tr valign="top">
<td height="541" bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: none; border-right: 1px solid #000000; padding-top: 0in; padding-bottom: 0in; padding-left: 0in; padding-right: 0.02in">
<p align="left"><font face="Arial"><font size="2" style="font-size: 10pt"><b>Generating
a Gaussian dataset:<br><br>creating random vectors from the
multivariate normal<br>distribution given mean and covariance
matrix<br><br>(here: 5 random vectors with <br>mean 0, covariance
= 0, variance = 2)</b></font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: none; border-right: 1px dotted #000000; padding-top: 0in; padding-bottom: 0in; padding-left: 0in; padding-right: 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt">%
requires statistics toolbox package<br>% how to install and load
it in Octave:<br><br>% download the package from:&nbsp;<br>%
http://octave.sourceforge.net/packages.php<br>% pkg install&nbsp;<br>%
&nbsp; &nbsp; ~/Desktop/io-2.0.2.tar.gz &nbsp;<br>% pkg install&nbsp;<br>%
&nbsp; &nbsp; ~/Desktop/statistics-1.2.3.tar.gz<br><br><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>
pkg load statistics<br><br><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>
mean = [0 0]<br><br><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>
cov = [2 0; 0 2]<br>cov =<br>&nbsp; &nbsp;2 &nbsp; 0<br>&nbsp; &nbsp;0
&nbsp; 2<br><br><font color="#006699"><font size="2" style="font-size: 10pt"><b>M&gt;</b></font></font>
mvnrnd(mean,cov,5)<br>&nbsp; &nbsp;2.480150 &nbsp;-0.559906<br>&nbsp;
-2.933047 &nbsp; 0.560212<br>&nbsp; &nbsp;0.098206 &nbsp;
3.055316<br>&nbsp; -0.985215 &nbsp;-0.990936<br>&nbsp; &nbsp;1.122528
&nbsp; 0.686977<br>&nbsp; &nbsp;&nbsp;</font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: 1px dotted #000000; padding: 0in 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt"><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
mean = np.array([0,0])<br><br><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
cov = np.array([[2,0],[0,2]])<br><br><font color="#ff420e"><font size="2" style="font-size: 10pt"><b>P&gt;</b></font></font>
np.random.multivariate_normal(mean, cov, 5)<br><br>Array([[
1.55432624, -1.17972629],&nbsp;<br>&nbsp; &nbsp; &nbsp;
&nbsp;[-2.01185294, 1.96081908],&nbsp;<br>&nbsp; &nbsp; &nbsp;
&nbsp;[-2.11810813, 1.45784216],&nbsp;<br>&nbsp; &nbsp; &nbsp;
&nbsp;[-2.93207591, -0.07369322],&nbsp;<br>&nbsp; &nbsp; &nbsp;
&nbsp;[-1.37031244, -1.18408792]])</font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: 1px dotted #000000; padding: 0in 0.02in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt">#
requires the ‘mass’ package<br><br>
<font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
install.packages('MASS')<br><br>
<font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
library(MASS)
<br><br><font color="#0000ff"><font size="2" style="font-size: 10pt"><b>R&gt;</b></font></font>
mvrnorm(n=10, mean, cov)<br>[,1] [,2]<br>[1,] -0.8407830
-0.1882706<br>[2,] 0.8496822 -0.7889329<br>[3,] -0.1564171
0.8422177<br>[4,] -0.6288779 1.0618688<br>[5,] -0.5103879
0.1303697<br>[6,] 0.8413189 -0.1623758<br>[7,] -1.0495466
-0.4161082<br>[8,] -1.3236339 0.7755572<br>[9,] 0.2771013
1.4900494<br>[10,] -1.3536268 0.2338913</font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: 1px dotted #000000; border-right: none; padding-top: 0in; padding-bottom: 0in; padding-left: 0.02in; padding-right: 0in">
<p align="left"><font face="Andale Mono"><font size="1" style="font-size: 8pt">#
requires the Distributions package from
https://github.com/JuliaStats/Distributions.jl<br><br><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>
using Distributions<br><br><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>
mean=[0., 0.]<br>2-element Array{Float64,1}:<br>0.0<br>0.0<br><br><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>
cov=[2. 0.; 0. 2.]<br>2x2 Array{Float64,2}:<br>2.0 0.0<br>0.0
2.0<br><br><font color="#9966cc"><font size="2" style="font-size: 10pt"><b>J&gt;</b></font></font>
rand( MvNormal(mean, cov), 5)<br>2x5 Array{Float64,2}:<br>-0.527634
0.370725 -0.761928 -3.91747 1.47516<br>-0.448821 2.21904 2.24561
0.692063 0.390495</font></font></p>
</td>
<td bgcolor="#ffffff" style="border-top: none; border-bottom: none; border-left: 1px solid #000000; border-right: 1px solid #000000; padding: 0in 0.02in">
<p align="left"><font face="Arial"><font size="2" style="font-size: 10pt"><b>Generating
a Gaussian dataset:<br><br>creating random vectors from the
multivariate normal<br>distribution given mean and covariance
matrix<br><br>(here: 5 random vectors with <br>mean 0, covariance
= 0, variance = 2)</b></font></font></p>
</td>
</tr>
</table>
<p><!-- ************************************************************************** --><br><br>
</p>
</body>
</html>
Reference in New Issue View Git Blame Copy Permalink