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<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>
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<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>
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<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>
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<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>
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<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>
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<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>
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<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>
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<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>
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<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>
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<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>
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<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>
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<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>
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<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>
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<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>
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<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>
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</table>
<p><!-- ************************************************************************** --><br><br>
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