diff --git a/tutorials/matrix_cheatsheet_only.html b/tutorials/matrix_cheatsheet_only.html index 7624ea4..c0d1a18 100644 --- a/tutorials/matrix_cheatsheet_only.html +++ b/tutorials/matrix_cheatsheet_only.html @@ -3,9 +3,9 @@ - + - + - - - - - - - +
+ + + + + + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
+

Task

 -

MATLAB/Octave

+ 		+

MATLAB/Octave

 -

Python +

+

Python NumPy

 -

R

+ 		+

R

 -

Julia

+ 		+

Julia

 +

Task
+

CREATING MATRICES
+

Creating Matrices 
(here: 3x3 matrix)

 -

M> +

+

M> A = [1 2 3; 4 5 6; 7 8 9]
A =
   1   2   3
   4   5   6
   7   8   9

 -

P> +

+

P> A = np.array([ [1,2,3], [4,5,6], [7,8,9] ])

P> A
array([[1, 2, 3],
       [4, 5, 6],
       [7, 8, 9]])

 -

R> +

+

R> A = matrix(c(1,2,3,4,5,6,7,8,9),nrow=3,byrow=T)


# equivalent to

# A = matrix(1:9,nrow=3,byrow=T)



R> A
[,1] [,2] [,3]
[1,] 1 2 3
[2,] 4 5 6
[3,] 7 8 9

 -

J> +

+

J> A=[1 2 3; 4 5 6; 7 8 9]
3x3 Array{Int64,2}:
1 2 3
4 5 6
7 8 9

 +

Creating Matrices 
(here: 3x3 matrix)
+

Creating an 1D column vector

 -

M> +

+

M> a = [1; 2; 3]
a =
   1
   2
   3

 -

P> - a - = +

+

P> + a + = np.array([1,2,3]).reshape(1,3)

-


P> - b.shape
(1, +


P> + b.shape
(1, 3)


 -

R> +

+

R> a = matrix(c(1,2,3), nrow=3, byrow=T)

R> a
[,1]
[1,] 1
[2,] 2
[3,] 3

 -

J> +

+

J> a=[1; 2; 3]
3-element Array{Int64,1}:
1
2
3

 +

Creating an 1D column vector
+

Creating an
1D row vector

 -

M> +

+

M> b = [1 2 3]
b =
   1   2   3

 -

P> +

+

P> b = np.array([1,2,3])

P> b
array([1, 2, 3])

-

# +

# note that numpy doesn't have
# explicit “row-vectors”, but 1-D
# arrays

-

P> - b.shape

-

(3,)

+

P> + b.shape

+

(3,)


 -

R> +

+

R> b = matrix(c(1,2,3), ncol=3)

R> b
[,1] [,2] [,3]
[1,] 1 2 3

 -

J> +

+

J> b=[1 2 3]
1x3 Array{Int64,2}:
1 2 3

# note that this is a 2D array.
# vectors in Julia are columns

 +

Creating an
1D row vector
+

Creating a
random m x n matrix

 -

M> +

+

M> rand(3,2)
ans =
   0.21977   0.10220
   0.38959   0.69911
   0.15624   0.65637

 -

P> +

+

P> np.random.rand(3,2)
array([[ 0.29347865,  0.17920462],
       [ 0.51615758,  0.64593471],
       [ 0.01067605,  0.09692771]])

 -

R> +

+

R> matrix(runif(3*2), ncol=2)
[,1] [,2]
[1,] 0.5675127 0.7751204
[2,] 0.3439412 0.5261893
[3,] 0.2273177 0.223438

 -

J> +

+

J> rand(3,2)
3x2 Array{Float64,2}:
0.36882 0.267725
0.571856 0.601524
0.848084 0.858935

 +

Creating a
random m x n matrix
+

Creating a
zero m x n matrix 

 -

M> +

+

M> zeros(3,2)
ans =
   0   0
   0   0
   0   0

 -

P> +

+

P> np.zeros((3,2))
array([[ 0.,  0.],
       [ 0.,  0.],
       [ 0.,  0.]])

 -

R> +

+

R> mat.or.vec(3, 2)
[,1] [,2]
[1,] 0 0
[2,] 0 0
[3,] 0 0

 -

J> +

+

J> zeros(3,2)
3x2 Array{Float64,2}:
0.0 0.0
0.0 0.0
0.0 0.0

 +

Creating a
zero m x n matrix 
+

Creating an
m x n matrix of ones

 -

M> +

+

M> ones(3,2)
ans =
   1   1
   1   1
   1   1

 -

P> +

+

P> np.ones((3,2))
array([[ 1.,  1.],
       [ 1.,  1.],
       [ 1.,  1.]])

 -

R> - mat.or.vec(3, 2) + 1
[,1] [,2]
[1,] 1 1
[2,] 1 1
[3,] - 1 1

+ 		+

R> + matrix(1L, + 3, 2)

[,1] + [,2]
[1,] 1 1
[2,] 1 1
[3,] 1 1

 -

J> +

+

J> ones(3,2)
3x2 Array{Float64,2}:
1.0 1.0
1.0 1.0
1.0 1.0

 +

Creating an
m x n matrix of ones
+

Creating an
identity matrix

 -

M> +

+

M> eye(3)
ans =
Diagonal Matrix
   1   0   0
   0   1   0
   0   0   1

 -

P> +

+

P> np.eye(3)
array([[ 1.,  0.,  0.],
       [ 0.,  1.,  0.],
       [ 0.,  0.,  1.]])

 -

R> +

+

R> diag(3)
[,1] [,2] [,3]
[1,] 1 0 0
[2,] 0 1 0
[3,] 0 0 1

 -

J> +

+

J> eye(3)
3x3 Array{Float64,2}:
1.0 0.0 0.0
0.0 1.0 0.0
0.0 0.0 1.0

 +

Creating an
identity matrix
+

Creating a
diagonal matrix

 -

M> +

+

M> a = [1 2 3]

M> diag(a)
ans =
Diagonal Matrix
   1   0   0
   0   2   0
   0   0   3

 -

P> +

+

P> a = np.array([1,2,3])

P> np.diag(a)
array([[1, 0, 0],
       [0, 2, 0],
       [0, 0, 3]])

 -

R> +

+

R> diag(1:3)
[,1] [,2] [,3]
[1,] 1 0 0
[2,] 0 2 0
[3,] 0 0 3

 -

J> +

+

J> a=[1, 2, 3]

# added commas because julia
# vectors are columnar

J> diagm(a)
3x3 Array{Int64,2}:
1 0 0
0 2 0
0 0 3

 +

Creating a
diagonal matrix
+

ACCESSING MATRIX ELEMENTS
+

Getting the dimension
of a matrix
(here: 2D, rows x cols)

 -

M> +

+

M> A = [1 2 3; 4 5 6]
A =
   1   2   3
   4   5   6

M> size(A)
ans =
   2   3

 -

P> +

+

P> A = np.array([ [1,2,3], [4,5,6] ])

P> A
array([[1, 2, 3],
       [4, 5, 6]])

P> A.shape
(2, 3)

 -

R> +

+

R> A = matrix(1:6,nrow=2,byrow=T)

R> A
[,1] [,2] [,3]
[1,] 1 2 3
[2,] 4 5 6


-

R> +

R> dim(A)
[1] 2 3

 -

J> +

+

J> A=[1 2 3; 4 5 6]
2x3 Array{Int64,2}:
1 2 3
4 5 6

J> size(A)
(2,3)

 +

Getting the dimension
of a matrix
(here: 2D, rows x cols)
+

Selecting rows 

 -

M> +

+

M> A = [1 2 3; 4 5 6; 7 8 9]

% 1st row
M> A(1,:)
ans =
   1   2   3

% 1st 2 rows
M> A(1:2,:)
ans =
   1   2   3
   4   5   6

 -

P> +

+

P> A = np.array([ [1,2,3], [4,5,6], [7,8,9] ])

# 1st row
P> A[0,:]
array([1, 2, 3])

# 1st 2 rows
P> A[0:2,:]
array([[1, 2, 3], [4, 5, 6]])

 -

R> +

+

R> A = matrix(1:9,nrow=3,byrow=T)



# 1st row


R> A[1,]
[1] 1 2 3



# 1st 2 rows


R> A[1:2,]
[,1] [,2] [,3]
[1,] 1 2 3
[2,] 4 5 6

 -

J> +

+

J> A=[1 2 3; 4 5 6; 7 8 9];
#semicolon suppresses output

#1st row
J> A[1,:]
1x3 Array{Int64,2}:
1 2 3

#1st 2 rows
J> A[1:2,:]
2x3 Array{Int64,2}:
1 2 3
4 5 6

 +

Selecting rows 
+

Selecting columns

 -

M> +

+

M> A = [1 2 3; 4 5 6; 7 8 9]

% 1st column
M> A(:,1)
ans =
   1
   4
   7

% 1st 2 columns
M> A(:,1:2)
ans =
   1   2
   4   5
   7   8

 -

P> +

+

P> A = np.array([ [1,2,3], [4,5,6], [7,8,9] ])

# 1st column (as row vector)
P> A[:,0]
array([1, 4, 7])

# 1st column (as column @@ -412,8 +413,8 @@ A[:,0:2]
array([[1, 2], 
       [4, 5], 
       [7, 8]])

 -

R> +

+

R> A = matrix(1:9,nrow=3,byrow=T)




# 1st column as row vector

R> t(A[,1])
[,1] [,2] [,3]
[1,] 1 4 7



# 1st column @@ -421,136 +422,193 @@ A[,1]
[1] 1 4 7



# 1st 2 columns

R> A[,1:2]
[,1] [,2]
[1,] 1 2
[2,] 4 5
[3,] 7 8

 -

J> +

+

J> A=[1 2 3; 4 5 6; 7 8 9];

#1st column
J> A[:,1]
3-element Array{Int64,1}:
1
4
7

#1st 2 columns
J> A[:,1:2]
3x2 Array{Int64,2}:
1 2
4 5
7 8

 +

Selecting columns
+

Extracting rows and columns by criteria

(here: get rows that have value 9 in column 3)

 -

M> +

+

M> A = [1 2 3; 4 5 9; 7 8 9]
A =
   1   2   3
   4   5   9
   7   8   9

M> A(A(:,3) == 9,:)
ans =
   4   5   9
   7   8   9

 -

P> +

+

P> A = np.array([ [1,2,3], [4,5,9], [7,8,9]])

P> A
array([[1, 2, 3],
       [4, 5, 9],
       [7, 8, 9]])

P> A[A[:,2] == 9]
array([[4, 5, 9],
       [7, 8, 9]])

 -

R> - A = matrix(1:9,nrow=3,byrow=T)



R> +

+

R> + A = matrix(1:9,nrow=3,byrow=T)



R> A
[,1] [,2] [,3]
[1,] 1 2 3
[2,] 4 5 9
[3,] 7 8 - 9



R> - matrix(A[A[,3]==9], ncol=3)
[,1] [,2] [,3]
[1,] 4 5 9
[2,] + 9



R> + A[A[,3]==9,]

+

[1] 7 8 9

+


+

-

J> +

+

J> A=[1 2 3; 4 5 9; 7 8 9]
3x3 Array{Int64,2}:
1 2 3
4 5 9
7 8 9

# use '.==' for
# element-wise check
J> A[ A[:,3] .==9, :]
2x3 Array{Int64,2}:
4 5 9
7 8 9

 +

Extracting rows and columns by criteria

(here: get rows that have value 9 in column 3)
+

Accessing elements
(here: 1st element)

 -

M> +

+

M> A = [1 2 3; 4 5 6; 7 8 9]

M> A(1,1)
ans =  1

 -

P> +

+

P> A = np.array([ [1,2,3], [4,5,6], [7,8,9] ])

P> A[0,0]
1

 -

R> +

+

R> A = matrix(c(1,2,3,4,5,9,7,8,9),nrow=3,byrow=T)



R> A[1,1]
[1] 1

 -

J> +

+

J> A=[1 2 3; 4 5 6; 7 8 9];

J> A[1,1]
1

 +

Accessing elements
(here: 1st element)
+

MANIPULATING SHAPE AND DIMENSIONS
+ +

Converting 
a + matrix into a row vector (by column)

+ 		+

M> + A = [1 2 3; 4 5 6; 7 8 9]

+

M> + A(:)

+

ans + = +

+

1 +
4
7
2
5
8
3
6
9

+ 		+

P> + A = np.array([[1,2,3],[4,5,6],[7,8,9]])

+

P> + A.flatten(1)

+

array([1, + 4, 7, 2, 5, 8, 3, 6, 9])

+


+

+ 		+

R> + A = matrix(1:9,nrow=3,byrow=T)

+


+

+

R> + as.vector(A)

+


+

+

[1] + 1 4 7 2 5 8 3 6 9

+


+

+ 		+

J> + A=[1 2 3; 4 5 6; 7 8 9]

+

J> + vec(A)

+

9-element + Array{Int64,1}:

+

1
4
7
2
5
8
3
6
9

+ 		+

Converting 
a + matrix into a row vector (by column)

+

Converting 
row to column vectors

 -

M> +

+

M> b = [1 2 3]


M> b = b'
b =
   1
   2
   3

 -

P> +

+

P> b = np.array([1, 2, 3])

P> b = b[np.newaxis].T
# alternatively
# b = b[:,np.newaxis]

P> b
array([[1],
       [2],
       [3]])

 -

R> +

+

R> b = matrix(c(1,2,3), ncol=3)

R> t(b)
[,1]
[1,] 1
[2,] 2
[3,] 3

 -

J> +

+

J> b=vec([1 2 3])
3-element Array{Int64,1}:
1
2
3

 +

Converting 
row to column vectors
+

Reshaping Matrices

(here: 3x3 matrix to row vector)

 -

M> +

+

M> A = [1 2 3; 4 5 6; 7 8 9]
A =
   1   2   3
   4   5   6
   7   8   9

M> @@ -559,19 +617,19 @@ =
   1   4   7   2   5   8   3   6   9

 -

P> +

+

P> A = np.array([[1,2,3],[4,5,6],[7,8,9]])

P> A
array([[1, 2, 3],
       [4, 5, 9],
       [7, 8, 9]])

P> total_elements = np.prod(A.shape)

-

P> +

P> B = A.reshape(1, total_elements) 

# alternative shortcut:
# A.reshape(1,-1)

P> B
array([[1, 2, 3, 4, 5, 6, 7, 8, 9]])

 -

R> +

+

R> A = matrix(1:9,nrow=3,byrow=T)



R> A
[,1] [,2] [,3]
[1,] 1 2 3
[2,] 4 5 6
[3,] 7 8 9


R> total_elements = dim(A)[1] * dim(A)[2]

R> @@ -579,33 +637,33 @@ B
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9]
[1,] 1 4 7 2 5 8 3 6 9

 -

J> +

+

J> A=[1 2 3; 4 5 6; 7 8 9]
3x3 Array{Int64,2}:
1 2 3
4 5 6
7 8 9

J> total_elements=length(A)
9

J>B=reshape(A,1,total_elements)
1x9 Array{Int64,2}:
1 4 7 2 5 8 3 6 9

 +

Reshaping Matrices

(here: 3x3 matrix to row vector)
+

Concatenating matrices

 -

M> +

+

M> A = [1 2 3; 4 5 6]

M> B = [7 8 9; 10 11 12]

M> C = [A; B]
    1    2    3
    4    5    6
    7    8    9
   10   11   12

 -

P> +

+

P> A = np.array([[1, 2, 3], [4, 5, 6]])

P> B = np.array([[7, 8, 9],[10,11,12]])

P> C = np.concatenate((A, B), axis=0)

P> @@ -613,33 +671,33 @@ 5, 6], 
       [ 7, 8, 9], 
       [10, 11, 12]])

 -

R> +

+

R> A = matrix(1:6,nrow=2,byrow=T)

R> B = matrix(7:12,nrow=2,byrow=T)

R> C = rbind(A,B)

R> C
[,1] [,2] [,3]
[1,] 1 2 3
[2,] 4 5 6
[3,] 7 8 9
[4,] 10 11 12

 -

J> +

+

J> A=[1 2 3; 4 5 6];

J> B=[7 8 9; 10 11 12];

J> C=[A; B]
4x3 Array{Int64,2}:
1 2 3
4 5 6
7 8 9
10 11 12

 +

Concatenating matrices
+

Stacking 
vectors and matrices

 -

M> +

+

M> a = [1 2 3]

M> b = [4 5 6]

M> c = [a' b']
c =
   1   4
   2   @@ -647,8 +705,8 @@ c = [a; b]
c =
   1   2   3
   4   5   6

 -

P> +

+

P> a = np.array([1,2,3])
P> b = np.array([4,5,6])

P> np.c_[a,b]
array([[1, 4],
       [2, @@ -656,38 +714,38 @@ np.r_[a,b]
array([[1, 2, 3],
       [4, 5, 6]])

 -

R> +

+

R> a = matrix(1:3, ncol=3)

R> b = matrix(4:6, ncol=3)

R> matrix(rbind(A, B), ncol=2)
[,1] [,2]
[1,] 1 5
[2,] 4 3



R> rbind(A,B)
[,1] [,2] [,3]
[1,] 1 2 3
[2,] 4 5 6

 -

J> +

+

J> a=[1 2 3];

J> b=[4 5 6];

J> c=[a' b']
3x2 Array{Int64,2}:
1 4
2 5
3 6

J> c=[a; b]
2x3 Array{Int64,2}:
1 2 3
4 5 6

 +

Stacking 
vectors and matrices
+

BASIC MATRIX OPERATIONS
+

Matrix-scalar
operations

 -

M> A +

+

M> A = [1 2 3; 4 5 6; 7 8 9]

M> A * 2
ans =
    2    4    6
    8   10   12
   14   16   @@ -696,30 +754,30 @@ A - 2

M> A / 2

 -

P> +

+

P> A = np.array([ [1,2,3], [4,5,6], [7,8,9] ])

P> A * 2
array([[ 2,  4,  6],
       [ 8, 10, 12],
       [14, 16, 18]])

P> A + 2

P> A - 2

P> A / 2

-

# +

# Note that NumPy was optimized for
# in-place assignments
# e.g., A += A instead of
# A = A + A

 -

R> +

+

R> A = matrix(1:9, nrow=3, byrow=T)

R> A * 2
[,1] [,2] [,3]
[1,] 2 4 6
[2,] 8 10 12
[3,] 14 16 18


-

R> +

R> A + 2

R> A - 2

R> A / 2

 -

J> +

+

J> A=[1 2 3; 4 5 6; 7 8 9];

# elementwise operator

J> A .* 2
3x3 Array{Int64,2}:
2 4 6
8 10 12
14 16 18

J> @@ -727,83 +785,83 @@ A .- 2;

J> A ./ 2;

 +

Matrix-scalar
operations
+

Matrix-matrix
multiplication

 -

M> A +

+

M> A = [1 2 3; 4 5 6; 7 8 9]

M> A * A
ans =
    30    36    42
    66    81    96
   102   126   150

 -

P> +

+

P> A = np.array([ [1,2,3], [4,5,6], [7,8,9] ])

P> np.dot(A,A) # or A.dot(A)
array([[ 30,  36,  42],
       [ 66,  81,  96],
       [102, 126, 150]])

 -

R> +

+

R> A = matrix(1:9, nrow=3, byrow=T)

R> A %*% A
[,1] [,2] [,3]
[1,] 30 36 42
[2,] 66 81 96
[3,] 102 126 150

 -

J> +

+

J> A=[1 2 3; 4 5 6; 7 8 9];

J> A * A
3x3 Array{Int64,2}:
30 36 42
66 81 96
102 126 150

 +

Matrix-matrix
multiplication
+

Matrix-vector
multiplication

 -

M> +

+

M> A = [1 2 3; 4 5 6; 7 8 9]

M> b = [ 1; 2; 3 ]

M> A * b
ans =
   14
   32
   50

 -

P> +

+

P> A = np.array([ [1,2,3], [4,5,6], [7,8,9] ])

P> b = np.array([ [1], [2], [3] ])

P> np.dot(A,b) # or A.dot(b)

array([[14], [32], [50]])

 -

R> +

+

R> A = matrix(1:9, ncol=3)

R> b = matrix(1:3, nrow=3)



R> t(b %*% A)
[,1]
[1,] 14
[2,] 32
[3,] 50

 -

J> +

+

J> A=[1 2 3; 4 5 6; 7 8 9];

J> b=[1; 2; 3];

J> A*b
3-element Array{Int64,1}:
14
32
50

 +

Matrix-vector
multiplication
+

Element-wise 
matrix-matrix operations

 -

M> A +

+

M> A = [1 2 3; 4 5 6; 7 8 9]

M> A .* A
ans =
    1    4    9
   16   25   36
   49   @@ -812,20 +870,20 @@ A .- A

M> A ./ A

 -

P> +

+

P> A = np.array([ [1,2,3], [4,5,6], [7,8,9] ])

P> A * A
array([[ 1,  4,  9],
       [16, 25, 36],
       [49, 64, 81]])

P> A + A

P> A - A

P> A / A

-

# +

# Note that NumPy was optimized for
# in-place assignments
# e.g., A += A instead of
# A = A + A

 -

R> +

+

R> A = matrix(1:9, nrow=3, byrow=T)


R> A * A
[,1] [,2] [,3]
[1,] 1 4 9
[2,] 16 25 36
[3,] 49 64 81



R> @@ -833,8 +891,8 @@ A - A

R> A / A

 -

J> +

+

J> A=[1 2 3; 4 5 6; 7 8 9];

J> A .* A
3x3 Array{Int64,2}:
1 4 9
16 25 36
49 64 81

J> @@ -842,67 +900,67 @@ A .- A;

J> A ./ A;

 +

Element-wise 
matrix-matrix operations
+

Matrix elements to power n

(here: individual elements squared)

 -

M> A +

+

M> A = [1 2 3; 4 5 6; 7 8 9]

M> A.^2
ans =
    1    4    9
   16   25   36
   49   64   81

 -

P> +

+

P> A = np.array([ [1,2,3], [4,5,6], [7,8,9] ])

P> np.power(A,2)
array([[ 1,  4,  9],
       [16, 25, 36],
       [49, 64, 81]])

 -

R> +

+

R> A = matrix(1:9, nrow=3, byrow=T)

R> A ^ 2
[,1] [,2] [,3]
[1,] 1 4 9
[2,] 16 25 36
[3,] 49 64 81

 -

J> +

+

J> A=[1 2 3; 4 5 6; 7 8 9];

J> A .^ 2
3x3 Array{Int64,2}:
1 4 9
16 25 36
49 64 81

 +

Matrix elements to power n

(here: individual elements squared)
+

Matrix to power n

(here: matrix-matrix multiplication with itself)

 -

M> A +

+

M> A = [1 2 3; 4 5 6; 7 8 9]

M> A ^ 2
ans =
    30    36    42
    66    81    96
   102   126   150

 -

P> +

+

P> A = np.array([ [1,2,3], [4,5,6], [7,8,9] ])

P> np.linalg.matrix_power(A,2)
array([[ 30,  36,  42],
       [ 66,  81,  96],
       [102, 126, 150]])

 -

R> +

+

R> A = matrix(1:9, ncol=3)


# requires the ‘expm’ package


R> install.packages('expm')


R> @@ -910,102 +968,102 @@ A %^% 2
[,1] [,2] [,3]
[1,] 30 66 102
[2,] 36 81 126
[3,] 42 96 150

 -

J> +

+

J> A=[1 2 3; 4 5 6; 7 8 9];

J> A ^ 2
3x3 Array{Int64,2}:
30 36 42
66 81 96
102 126 150

 +

Matrix to power n

(here: matrix-matrix multiplication with itself)
+

Matrix transpose

 -

M> A +

+

M> A = [1 2 3; 4 5 6; 7 8 9]

M> A'
ans =
   1   4   7
   2   5   8
   3   6   9

 -

P> +

+

P> A = np.array([ [1,2,3], [4,5,6], [7,8,9] ])

P> A.T
array([[1, 4, 7],
       [2, 5, 8],
       [3, 6, 9]])

 -

R> +

+

R> A = matrix(1:9, nrow=3, byrow=T)


R> t(A)
[,1] [,2] [,3]
[1,] 1 4 7
[2,] 2 5 8
[3,] 3 6 9

 -

J> +

+

J> A=[1 2 3; 4 5 6; 7 8 9]
3x3 Array{Int64,2}:
1 2 3
4 5 6
7 8 9

J> A'
3x3 Array{Int64,2}:
1 4 7
2 5 8
3 6 9

 +

Matrix transpose
+

Determinant of a matrix:
 A -> |A|

 -

M> +

+

M> A = [6 1 1; 4 -2 5; 2 8 7]
A =
   6   1   1
   4  -2   5
   2   8   7

M> det(A)
ans = -306

 -

P> A +

+

P> A = np.array([[6,1,1],[4,-2,5],[2,8,7]])

P> A
array([[ 6,  1,  1],
       [ 4, -2,  5],
       [ 2,  8,  7]])

P> np.linalg.det(A)
-306.0

 -

R> +

+

R> A = matrix(c(6,1,1,4,-2,5,2,8,7), nrow=3, byrow=T)

R> A
[,1] [,2] [,3]
[1,] 6 1 1
[2,] 4 -2 5
[3,] 2 8 7

R> det(A)
[1] -306

 -

J> +

+

J> A=[6 1 1; 4 -2 5; 2 8 7]
3x3 Array{Int64,2}:
6 1 1
4 -2 5
2 8 7

J> det(A)
-306.0

 +

Determinant of a matrix:
 A -> |A|
+

Inverse of a matrix

 -

M> +

+

M> A = [4 7; 2 6]
A =
   4   7
   2   6

M> A_inv = inv(A)
A_inv =
   0.60000  -0.70000
  -0.20000   0.40000

 -

P> +

+

P> A = np.array([[4, 7], [2, 6]])

P> A
array([[4, 7], 
       [2, 6]])

P> @@ -1013,36 +1071,36 @@ A_inverse
array([[ 0.6, -0.7], 
       [-0.2, 0.4]])

 -

R> +

+

R> A = matrix(c(4,7,2,6), nrow=2, byrow=T)

R> A
[,1] [,2]
[1,] 4 7
[2,] 2 6

R> solve(A)
[,1] [,2]
[1,] 0.6 -0.7
[2,] -0.2 0.4

 -

J> +

+

J> A=[4 7; 2 6]
2x2 Array{Int64,2}:
4 7
2 6

J> A_inv=inv(A)
2x2 Array{Float64,2}:
0.6 -0.7
-0.2 0.4

 +

Inverse of a matrix
+

ADVANCED MATRIX OPERATIONS
+

Calculating the covariance matrix 
of 3 random variables

(here: covariances of the means 
of x1, x2, and x3)

 -

M> +

+

M> x1 = [4.0000 4.2000 3.9000 4.3000 4.1000]’

M> x2 = [2.0000 2.1000 2.0000 2.1000 2.2000]'

M> x3 = [0.60000 0.59000 0.58000 0.62000 0.63000]’

M> @@ -1051,8 +1109,8 @@ 7.0000e-03   1.3500e-03
   1.7500e-03   1.3500e-03   4.3000e-04

 -

P> +

+

P> x1 = np.array([ 4, 4.2, 3.9, 4.3, 4.1])

P> x2 = np.array([ 2, 2.1, 2, 2.1, 2.2])

P> x3 = np.array([ 0.6, 0.59, 0.58, 0.62, 0.63])

P> @@ -1061,8 +1119,8 @@  ,  0.00135],
       [ 0.00175,  0.00135,  0.00043]])

 -

R> +

+

R> x1 = matrix(c(4, 4.2, 3.9, 4.3, 4.1), ncol=5)

R> x2 = matrix(c(2, 2.1, 2, 2.1, 2.2), ncol=5)

R> x3 = matrix(c(0.6, 0.59, 0.58, 0.62, 0.63), ncol=5)



R> @@ -1070,27 +1128,27 @@ 0.02500 0.00750 0.00175
[2,] 0.00750 0.00700 0.00135
[3,] 0.00175 0.00135 0.00043

 -

J> +

+

J> x1=[4.0 4.2 3.9 4.3 4.1]';

J> x2=[2. 2.1 2. 2.1 2.2]';

J> x3=[0.6 .59 .58 .62 .63]';

J> cov([x1 x2 x3])
3x3 Array{Float64,2}:
0.025 0.0075 0.00175
0.0075 0.007 0.00135
0.00175 0.00135 0.00043

 +

Calculating the covariance matrix 
of 3 random variables

(here: covariances of the means 
of x1, x2, and x3)
+

Calculating 
eigenvectors and eigenvalues

 -

M> +

+

M> A = [3 1; 1 3]
A =
   3   1
   1   3

M> [eig_vec,eig_val] = eig(A)
eig_vec =
  -0.70711   @@ -1098,8 +1156,8 @@ =
Diagonal Matrix
   2   0
   0   4

 -

P> +

+

P> A = np.array([[3, 1], [1, 3]])

P> A
array([[3, 1],
       [1, 3]])

P> eig_val, eig_vec = np.linalg.eig(A)

P> @@ -1107,34 +1165,34 @@ eig_vec
Array([[ 0.70710678, -0.70710678],
       [ 0.70710678,  0.70710678]])

 -

R> +

+

R> A = matrix(c(3,1,1,3), ncol=2)

R> A
[,1] [,2]
[1,] 3 1
[2,] 1 3

R> eigen(A)
$values
[1] 4 2

$vectors
[,1] [,2]
[1,] 0.7071068 -0.7071068
[2,] 0.7071068 0.7071068

 -

J> +

+

J> A=[3 1; 1 3]
2x2 Array{Int64,2}:
3 1
1 3

J> (eig_vec,eig_val)=eig(a)
([2.0,4.0],
2x2 Array{Float64,2}:
-0.707107 0.707107
0.707107 0.707107)

 +

Calculating 
eigenvectors and eigenvalues
+

Generating a Gaussian dataset:

creating random vectors from the multivariate normal
distribution given mean and covariance matrix

(here: 5 random vectors with
mean 0, covariance = 0, variance = 2)

 -

% +

+

% requires statistics toolbox package
% how to install and load it in Octave:

% download the package from: 
% http://octave.sourceforge.net/packages.php
% pkg install 
% @@ -1149,8 +1207,8 @@ 3.055316
  -0.985215  -0.990936
   1.122528   0.686977
    

 -

P> +

+

P> mean = np.array([0,0])

P> cov = np.array([[2,0],[0,2]])

P> np.random.multivariate_normal(mean, cov, 5)

Array([[ @@ -1160,8 +1218,8 @@  [-2.93207591, -0.07369322], 
       [-1.37031244, -1.18408792]])

 -

# +

+

# requires the ‘mass’ package


R> install.packages('MASS')


R> library(MASS)


R> @@ -1172,8 +1230,8 @@ -0.4161082
[8,] -1.3236339 0.7755572
[9,] 0.2771013 1.4900494
[10,] -1.3536268 0.2338913

 -

# +

+

# requires the Distributions package from https://github.com/JuliaStats/Distributions.jl

J> using Distributions

J> @@ -1184,7 +1242,7 @@ 0.370725 -0.761928 -3.91747 1.47516
-0.448821 2.21904 2.24561 0.692063 0.390495

 +

Generating a Gaussian dataset:

creating random vectors from the multivariate normal
distribution given mean and covariance