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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">

<html>

<head>

<!--

td p { color: #000000; font-family: "Verdana"; font-size: 12pt }

p { color: #000000; font-family: "Verdana"; font-size: 12pt }

-->

</style>

</head>

<tr>

Task

</td>

MATLAB/Octave

</td>

Python

NumPy

</td>

R

</td>

Julia

</td>

Task

</td>

</tr>

<tr>

CREATING

MATRICES

</td>

</tr>

Creating

Matrices  (here: 3x3 matrix)

</td>

M>

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

3    4   5   6    7   8

9

</td>

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]])

</td>

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

</td>

J>

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

8 9

</td>

Creating

Matrices  (here: 3x3 matrix)

</td>

</tr>

Creating

an 1D column vector

</td>

M>

a = [1; 2; 3] a =    1    2

3

</td>

P>

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

a array([[1],        [2],

[3]])

</td>

R>

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

a [,1] [1,] 1 [2,] 2 [3,] 3

</td>

J>

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

</td>

Creating

an 1D column vector

</td>

</tr>

Creating

an 1D row vector

</td>

M>

b = [1 2 3] b =    1   2   3

</td>

P>

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

b array([1, 2, 3])

</td>

R>

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

b [,1] [,2] [,3] [1,] 1 2 3

</td>

J>

b=[1 2 3] 1x3 Array{Int64,2}: 1 2 3 # note that this

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

</td>

Creating

an 1D row vector

</td>

</tr>

Creating

a random m x n matrix

</td>

M>

rand(3,2) ans =    0.21977   0.10220

0.38959   0.69911    0.15624   0.65637

</td>

P>

np.random.rand(3,2) array([[ 0.29347865,  0.17920462],

[ 0.51615758,  0.64593471],

[ 0.01067605,  0.09692771]])

</td>

R>

matrix(runif(3*2), ncol=2) [,1] [,2] [1,] 0.5675127

0.7751204 [2,] 0.3439412 0.5261893 [3,] 0.2273177 0.223438

</td>

J>

rand(3,2) 3x2 Array{Float64,2}: 0.36882 0.267725 0.571856

0.601524 0.848084 0.858935

</td>

Creating

a random m x n matrix

</td>

</tr>

Creating

a zero m x n matrix

</td>

M>

zeros(3,2) ans =    0   0    0

0    0   0

</td>

P>

np.zeros((3,2)) array([[ 0.,  0.],

[ 0.,  0.],        [ 0.,

0.]])

</td>

R>

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

</td>

J>

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

0.0

</td>

Creating

a zero m x n matrix

</td>

</tr>

Creating

an m x n matrix of ones

</td>

M>

ones(3,2) ans =    1   1    1

1    1   1

</td>

P>

np.ones([3,2]) array([[ 1.,  1.],

[ 1.,  1.],        [ 1.,  1.]])

</td>

R>

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

1 1

</td>

J>

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

1.0

</td>

Creating

an m x n matrix of ones

</td>

</tr>

Creating

an identity matrix

</td>

M>

eye(3) ans = Diagonal Matrix    1   0

0    0   1   0    0   0

1

</td>

P>

np.eye(3) array([[ 1.,  0.,  0.],

[ 0.,  1.,  0.],        [

0.,  0.,  1.]])

</td>

R>

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

1

</td>

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

</td>

Creating

an identity matrix

</td>

</tr>

Creating

a diagonal matrix

</td>

M>

a = [1 2 3] M>

diag(a) ans = Diagonal Matrix    1   0

0    0   2   0    0   0

3

</td>

P>

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

np.diag(a) array([[1, 0, 0],        [0,

2, 0],        [0, 0, 3]])

</td>

R>

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

0 3

</td>

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

</td>

Creating

a diagonal matrix

</td>

</tr>

<tr>

ACCESSING

MATRIX ELEMENTS

</td>

</tr>

Getting

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

</td>

M>

A = [1 2 3; 4 5 6] A =    1   2   3

4   5   6 M>

size(A) ans =    2   3

</td>

P>

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

A array([[1, 2, 3],        [4, 5,

6]]) P>

A.shape (2, 3)

</td>

R>

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

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

R>

dim(A) [1] 2 3

</td>

J>

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

size(A) (2,3)

</td>

Getting

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

</td>

</tr>

Selecting

rows

</td>

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

</td>

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]])

</td>

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

</td>

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

</td>

Selecting

rows

</td>

</tr>

Selecting

columns

</td>

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

</td>

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

vector) P>

A[:,[0]] array([[1],        [4],

[7]]) # 1st 2 columns P>

A[:,0:2] array([[1, 2],         [4,

5],         [7, 8]])

</td>

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

as column vector  R>

A[,1] [1] 1 4 7   # 1st 2 columns  R>

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

</td>

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

</td>

Selecting

columns

</td>

</tr>

Extracting

rows and columns by criteria (here: get rows that have

value 9 in column 3)

</td>

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

</td>

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]])

</td>

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,]

7 8 9

</td>

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

</td>

Extracting

rows and columns by criteria (here: get rows that have

value 9 in column 3)

</td>

</tr>

Accessing

elements (here: 1st element)

</td>

M>

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

A(1,1) ans =  1

</td>

P>

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

A[0,0] 1

</td>

R>

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

A[1,1] [1] 1

</td>

J>

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

A[1,1] 1

</td>

Accessing

elements (here: 1st element)

</td>

</tr>

<tr>

MANIPULATING

SHAPE AND DIMENSIONS

</td>

</tr>

Converting  row

to column vectors

</td>

M>

b = [1 2 3]  M>

b = b' b =    1    2

3

</td>

P>

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

b = b[np.newaxis].T # alternatively # b =

b[:,np.newaxis] P>

b array([[1],        [2],

[3]])

</td>

R>

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

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

</td>

J>

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

</td>

Converting  row

to column vectors

</td>

</tr>

Reshaping

Matrices (here: 3x3 matrix to row vector)

</td>

M>

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

3    4   5   6    7   8

9 M>

total_elements = numel(A) M>

B = reshape(A,1,total_elements)  % or reshape(A,1,9) B

=    1   4   7   2   5   8

3   6   9

</td>

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 = A.shape[0] * A.shape[1] P>

B = A.reshape(1, total_elements)  # or

A.reshape(1,9) # Alternative: A.shape = (1,9)  #

to change the array in place P>

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

</td>

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>

B = matrix(A, ncol=total_elements) R>

B [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [1,] 1 4 7 2

5 8 3 6 9

</td>

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

</td>

Reshaping

Matrices (here: 3x3 matrix to row vector)

</td>

</tr>

Concatenating

matrices

</td>

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

</td>

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>

C array([[ 1, 2, 3],         [ 4,

5, 6],         [ 7, 8, 9],

[10, 11, 12]])

</td>

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

</td>

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

</td>

Concatenating

matrices

</td>

</tr>

Stacking  vectors

and matrices

</td>

M>

a = [1 2 3] M>

b = [4 5 6] M>

c = [a' b'] c =    1   4    2

5    3   6 M>

c = [a; b] c =    1   2   3

4   5   6

</td>

P>

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

b = np.array([4,5,6]) P>

np.c_[a,b] array([[1, 4],        [2,

5],        [3, 6]]) P>

np.r_[a,b] array([[1, 2, 3],        [4,

5, 6]])

</td>

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

</td>

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

</td>

Stacking  vectors

and matrices

</td>

</tr>

<tr>

BASIC

MATRIX OPERATIONS

</td>

</tr>

Matrix-scalar operations

</td>

M> A

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

A * 2 ans =     2    4    6

8   10   12    14   16

18 M>

A + 2 M>

A - 2 M>

A / 2

</td>

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

</td>

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>

A + 2 R>

A - 2 R>

A / 2

</td>

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>

A .+ 2; J>

A .- 2; J>

A ./ 2;

</td>

Matrix-scalar operations

</td>

</tr>

Matrix-matrix multiplication

</td>

M> A

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

A * A ans =     30    36

42     66    81    96

102   126   150

</td>

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]])

</td>

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

</td>

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

</td>

Matrix-matrix multiplication

</td>

</tr>

Matrix-vector multiplication

</td>

M>

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

b = [ 1; 2; 3 ] M>

A * b ans =    14    32

50

</td>

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]])

</td>

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

</td>

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

</td>

Matrix-vector multiplication

</td>

</tr>

Element-wise  matrix-matrix operations

</td>

M> A

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

A .* A ans =     1    4

9    16   25   36    49

64   81 M>

A .+ A M>

A .- A M>

A ./ A

</td>

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

</td>

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>

A + A R>

A - A R>

A / A

</td>

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>

A .+ A; J>

A .- A; J>

A ./ A;

</td>

Element-wise  matrix-matrix operations

</td>

</tr>

Matrix

elements to power n (here: individual elements

squared)

</td>

M> A

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

A.^2 ans =     1    4    9

16   25   36    49   64   81

</td>

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]])

</td>

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

</td>

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

</td>

Matrix

elements to power n (here: individual elements

squared)

</td>

</tr>

Matrix

to power n (here: matrix-matrix multiplication with

itself)

</td>

M> A

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

A ^ 2 ans =     30    36

42     66    81    96

102   126   150

</td>

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]])

</td>

R>

A = matrix(1:9, ncol=3)  # requires the ‘expm’

package  R>

install.packages('expm')  R>

library(expm)   R>

A %^% 2 [,1] [,2] [,3] [1,] 30 66 102 [2,] 36 81 126 [3,]

42 96 150

</td>

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

</td>

Matrix

to power n (here: matrix-matrix multiplication with

itself)

</td>

</tr>

Matrix

transpose

</td>

M> A

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

A' ans =    1   4   7    2

5   8    3   6   9

</td>

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]])

</td>

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

</td>

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

</td>

Matrix

transpose

</td>

</tr>

Determinant

of a matrix:  A -> |A|

</td>

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

</td>

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

</td>

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

</td>

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

</td>

Determinant

of a matrix:  A -> |A|

</td>

</tr>

Inverse

of a matrix

</td>

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

</td>

P>

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

A array([[4, 7],         [2,

6]]) P>

A_inverse = np.linalg.inv(A) P>

A_inverse array([[ 0.6, -0.7],

[-0.2, 0.4]])

</td>

R>

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

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

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