| Task | -MATLAB/Octave | -Python NumPy | -R | -Julia | -Task | -||||||
| CREATING MATRICES [back to cheat sheet overview] |
- |||||||||||
| Creating Matrices (here: 3x3 matrix) |
- M> A = [1 2 3; 4 5 6; 7 8 9] A = 1 2 3 4 5 6 7 8 9 |
- 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> 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> 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) |
+
+ Task + |
+
+ MATLAB/Octave + |
+
+ Python + NumPy + |
+
+ R + |
+
+ Julia + |
+
+ Task + |
| Creating an 1D column vector | -M> a = [1; 2; 3] a = 1 2 3 |
- P> a = np.array([[1],[2],[3]]) P> a array([[1], [2], [3]]) |
- R> a = matrix(c(1,2,3), nrow=3, byrow=T) R> a [,1] [1,] 1 [2,] 2 [3,] 3 R> |
- J> a=[1; 2; 3] 3-element Array{Int64,1}: 1 2 3 |
- Creating an 1D column vector | +
+ CREATING
+ MATRICES |
+ |||||
|
+ Creating
+ Matrices |
+
+ M>
+ A = [1 2 3; 4 5 6; 7 8 9] |
+
+ P>
+ A = np.array([ [1,2,3], [4,5,6], [7,8,9] ]) |
+
+ R>
+ A = matrix(c(1,2,3,4,5,6,7,8,9),nrow=3,byrow=T) |
+
+ J>
+ A=[1 2 3; 4 5 6; 7 8 9] |
+
+ Creating
+ Matrices |
+ ||||||
|
+ Creating + an 1D column vector + |
+
+ M>
+ a = [1; 2; 3] |
+
+ P>
+ a = np.array([[1],[2],[3]]) |
+
+ R>
+ a = matrix(c(1,2,3), nrow=3, byrow=T) |
+
+ J>
+ a=[1; 2; 3] |
+
+ Creating + an 1D column vector + |
+ ||||||
|
+ Creating
+ an |
+
+ M>
+ b = [1 2 3] |
+
+ P>
+ b = np.array([1,2,3]) |
+
+ R>
+ b = matrix(c(1,2,3), ncol=3) |
+
+ J>
+ b=[1 2 3] |
+
+ Creating
+ an |
+ ||||||
|
+ Creating
+ a |
+
+ M>
+ rand(3,2) |
+
+ P>
+ np.random.rand(3,2) |
+
+ R>
+ matrix(runif(3*2), ncol=2) |
+
+ J>
+ rand(3,2) |
+
+ Creating
+ a |
+ ||||||
|
+ Creating
+ a |
+
+ M>
+ zeros(3,2) |
+
+ P>
+ np.zeros((3,2)) |
+
+ R>
+ mat.or.vec(3, 2) |
+
+ J>
+ zeros(3,2) |
+
+ Creating
+ a |
+ ||||||
|
+ Creating
+ an |
+
+ M>
+ ones(3,2) |
+
+ P>
+ np.ones([3,2]) |
+
+ R>
+ mat.or.vec(3, 2) + 1 |
+
+ J>
+ ones(3,2) |
+
+ Creating
+ an |
+ ||||||
|
+ Creating
+ an |
+
+ M>
+ eye(3) |
+
+ P>
+ np.eye(3) |
+
+ R>
+ diag(3) |
+
+ J>
+ eye(3) |
+
+ Creating
+ an |
+ ||||||
|
+ Creating
+ a |
+
+ M>
+ a = [1 2 3] |
+
+ P>
+ a = np.array([1,2,3]) |
+
+ R>
+ diag(1:3) |
+
+ J>
+ a=[1, 2, 3] |
+
+ Creating
+ a |
||||||
| Creating an 1D row vector |
- M> b = [1 2 3] b = 1 2 3 |
- P> b = np.array([1,2,3]) P> b array([1, 2, 3]) |
- R> b = matrix(c(1,2,3), ncol=3) R> b [,1] [,2] [,3] [1,] 1 2 3 |
- 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 |
+
+ ACCESSING
+ MATRIX ELEMENTS |
+ |||||
|
+ Getting
+ the dimension |
+
+ M>
+ A = [1 2 3; 4 5 6] |
+
+ P>
+ A = np.array([ [1,2,3], [4,5,6] ]) |
+
+ R>
+ A = matrix(1:6,nrow=2,byrow=T) R>
+ dim(A) |
+
+ J>
+ A=[1 2 3; 4 5 6] |
+
+ Getting
+ the dimension |
+ ||||||
|
+ Selecting + rows + |
+
+ M>
+ A = [1 2 3; 4 5 6; 7 8 9] |
+
+ P>
+ A = np.array([ [1,2,3], [4,5,6], [7,8,9] ]) |
+
+ R>
+ A = matrix(1:9,nrow=3,byrow=T)
|
+
+ J>
+ A=[1 2 3; 4 5 6; 7 8 9]; |
+
+ Selecting + rows + |
+ ||||||
|
+ Selecting + columns + |
+
+ M>
+ A = [1 2 3; 4 5 6; 7 8 9] |
+
+ P>
+ A = np.array([ [1,2,3], [4,5,6], [7,8,9] ]) |
+
+ R>
+ A = matrix(1:9,nrow=3,byrow=T)
|
+
+ J>
+ A=[1 2 3; 4 5 6; 7 8 9]; |
+
+ Selecting + columns + |
+ ||||||
|
+ Extracting
+ rows and columns by criteria |
+
+ M>
+ A = [1 2 3; 4 5 9; 7 8 9] |
+
+ P>
+ A = np.array([ [1,2,3], [4,5,9], [7,8,9]]) |
+
+ R>
+ A = matrix(1:9,nrow=3,byrow=T)
|
+
+ J>
+ A=[1 2 3; 4 5 9; 7 8 9] |
+
+ Extracting
+ rows and columns by criteria |
+ ||||||
|
+ Accessing
+ elements |
+
+ M>
+ A = [1 2 3; 4 5 6; 7 8 9] |
+
+ P>
+ A = np.array([ [1,2,3], [4,5,6], [7,8,9] ]) |
+
+ R>
+ A = matrix(c(1,2,3,4,5,9,7,8,9),nrow=3,byrow=T)
|
+
+ J>
+ A=[1 2 3; 4 5 6; 7 8 9]; |
+
+ Accessing
+ elements |
||||||
| Creating a random m x n matrix |
- M> rand(3,2) ans = 0.21977 0.10220 0.38959 0.69911 0.15624 0.65637 |
- P> np.random.rand(3,2) array([[ 0.29347865, 0.17920462], [ 0.51615758, 0.64593471], [ 0.01067605, 0.09692771]]) |
- 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> 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 |
+
+ MANIPULATING
+ SHAPE AND DIMENSIONS |
+ |||||
|
+ Converting |
+
+ M>
+ b = [1 2 3]
|
+
+ P>
+ b = np.array([1, 2, 3]) |
+
+ R>
+ b = matrix(c(1,2,3), ncol=3) |
+
+ J>
+ b=vec([1 2 3]) |
+
+ Converting |
+ ||||||
|
+ Reshaping
+ Matrices |
+
+ M>
+ A = [1 2 3; 4 5 6; 7 8 9] |
+
+ P>
+ A = np.array([[1,2,3],[4,5,6],[7,8,9]]) |
+
+ R>
+ A = matrix(1:9,nrow=3,byrow=T)
|
+
+ J>
+ A=[1 2 3; 4 5 6; 7 8 9] |
+
+ Reshaping
+ Matrices |
+ ||||||
|
+ Concatenating + matrices + |
+
+ M>
+ A = [1 2 3; 4 5 6] |
+
+ P>
+ A = np.array([[1, 2, 3], [4, 5, 6]]) |
+
+ R>
+ A = matrix(1:6,nrow=2,byrow=T) |
+
+ J>
+ A=[1 2 3; 4 5 6]; |
+
+ Concatenating + matrices + |
+ ||||||
|
+ Stacking |
+
+ M>
+ a = [1 2 3] |
+
+ P>
+ a = np.array([1,2,3]) |
+
+ R>
+ a = matrix(1:3, ncol=3) |
+
+ J>
+ a=[1 2 3]; |
+
+ Stacking |
||||||
| Creating a zero m x n matrix |
- M> zeros(3,2) ans = 0 0 0 0 0 0 |
- P> np.zeros((3,2)) array([[ 0., 0.], [ 0., 0.], [ 0., 0.]]) |
- R> mat.or.vec(3, 2) [,1] [,2] [1,] 0 0 [2,] 0 0 [3,] 0 0 |
- 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 |
+
+ BASIC
+ MATRIX OPERATIONS |
+ |||||
|
+ Matrix-scalar |
+
+ M> A
+ = [1 2 3; 4 5 6; 7 8 9] |
+
+ P>
+ A = np.array([ [1,2,3], [4,5,6], [7,8,9] ]) |
+
+ R>
+ A = matrix(1:9, nrow=3, byrow=T) R>
+ A + 2 |
+
+ J>
+ A=[1 2 3; 4 5 6; 7 8 9]; |
+
+ Matrix-scalar |
+ ||||||
|
+ Matrix-matrix |
+
+ M> A
+ = [1 2 3; 4 5 6; 7 8 9] |
+
+ P>
+ A = np.array([ [1,2,3], [4,5,6], [7,8,9] ]) |
+
+ R>
+ A = matrix(1:9, nrow=3, byrow=T) |
+
+ J>
+ A=[1 2 3; 4 5 6; 7 8 9]; |
+
+ Matrix-matrix |
+ ||||||
|
+ Matrix-vector |
+
+ M>
+ A = [1 2 3; 4 5 6; 7 8 9] |
+
+ P>
+ A = np.array([ [1,2,3], [4,5,6], [7,8,9] ]) |
+
+ R>
+ A = matrix(1:9, ncol=3) |
+
+ J>
+ A=[1 2 3; 4 5 6; 7 8 9]; |
+
+ Matrix-vector |
+ ||||||
|
+ Element-wise |
+
+ M> A
+ = [1 2 3; 4 5 6; 7 8 9] |
+
+ P>
+ A = np.array([ [1,2,3], [4,5,6], [7,8,9] ]) |
+
+ R>
+ A = matrix(1:9, nrow=3, byrow=T)
|
+
+ J>
+ A=[1 2 3; 4 5 6; 7 8 9]; |
+
+ Element-wise |
+ ||||||
|
+ Matrix
+ elements to power n |
+
+ M> A
+ = [1 2 3; 4 5 6; 7 8 9] |
+
+ P>
+ A = np.array([ [1,2,3], [4,5,6], [7,8,9] ]) |
+
+ R>
+ A = matrix(1:9, nrow=3, byrow=T) |
+
+ J>
+ A=[1 2 3; 4 5 6; 7 8 9]; |
+
+ Matrix
+ elements to power n |
+ ||||||
|
+ Matrix
+ to power n |
+
+ M> A
+ = [1 2 3; 4 5 6; 7 8 9] |
+
+ P>
+ A = np.array([ [1,2,3], [4,5,6], [7,8,9] ]) |
+
+ R>
+ A = matrix(1:9, ncol=3) |
+
+ J>
+ A=[1 2 3; 4 5 6; 7 8 9]; |
+
+ Matrix
+ to power n |
+ ||||||
|
+ Matrix + transpose + |
+
+ M> A
+ = [1 2 3; 4 5 6; 7 8 9] |
+
+ P>
+ A = np.array([ [1,2,3], [4,5,6], [7,8,9] ]) |
+
+ R>
+ A = matrix(1:9, nrow=3, byrow=T)
|
+
+ J>
+ A=[1 2 3; 4 5 6; 7 8 9] |
+
+ Matrix + transpose + |
+ ||||||
|
+ Determinant
+ of a matrix: |
+
+ M>
+ A = [6 1 1; 4 -2 5; 2 8 7] |
+
+ P> A
+ = np.array([[6,1,1],[4,-2,5],[2,8,7]]) |
+
+ R>
+ A = matrix(c(6,1,1,4,-2,5,2,8,7), nrow=3, byrow=T) |
+
+ J>
+ A=[6 1 1; 4 -2 5; 2 8 7] |
+
+ Determinant
+ of a matrix: |
+ ||||||
|
+ Inverse + of a matrix + |
+
+ M>
+ A = [4 7; 2 6] |
+
+ P>
+ A = np.array([[4, 7], [2, 6]]) |
+
+ R>
+ A = matrix(c(4,7,2,6), nrow=2, byrow=T) |
+
+ J>
+ A=[4 7; 2 6] |
+
+ Inverse + of a matrix + |
||||||
| Creating an m x n matrix of ones |
- M> ones(3,2) ans = 1 1 1 1 1 1 |
- 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 |
- 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 |
+
+ ADVANCED
+ MATRIX OPERATIONS |
|||||
| Creating an identity matrix |
- M> eye(3) ans = Diagonal Matrix 1 0 0 0 1 0 0 0 1 |
- P> np.eye(3) array([[ 1., 0., 0.], [ 0., 1., 0.], [ 0., 0., 1.]]) |
- R> diag(3) [,1] [,2] [,3] [1,] 1 0 0 [2,] 0 1 0 [3,] 0 0 1 |
- 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 |
+ ||||||
|
+ Calculating
+ the covariance matrix |
+
+ M>
+ x1 = [4.0000 4.2000 3.9000 4.3000 4.1000]’ |
+
+ P>
+ x1 = np.array([ 4, 4.2, 3.9, 4.3, 4.1]) |
+
+ R>
+ x1 = matrix(c(4, 4.2, 3.9, 4.3, 4.1), ncol=5) |
+
+ J>
+ x1=[4.0 4.2 3.9 4.3 4.1]'; |
+
+ Calculating
+ the covariance matrix |
||||||
| Creating a diagonal matrix |
- M> a = [1 2 3] M> diag(a) ans = Diagonal Matrix 1 0 0 0 2 0 0 0 3 |
- P> a = np.array([1,2,3]) P> np.diag(a) array([[1, 0, 0], [0, 2, 0], [0, 0, 3]]) |
- R> diag(1:3) [,1] [,2] [,3] [1,] 1 0 0 [2,] 0 2 0 [3,] 0 0 3 |
- 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 |
+ ||||||
|
+ Calculating |
+
+ M>
+ A = [3 1; 1 3] |
+
+ P>
+ A = np.array([[3, 1], [1, 3]]) |
+
+ R>
+ A = matrix(c(3,1,1,3), ncol=2) |
+
+ J>
+ A=[3 1; 1 3] |
+
+ Calculating |
||||||
| ACESSING MATRIX ELEMENTS | -|||||||||||
| Getting the dimension of a matrix (here: 2D, rows x cols) |
- M> A = [1 2 3; 4 5 6] A = 1 2 3 4 5 6 M> size(A) ans = 2 3 |
- 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> 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 |
- 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> 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> 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> 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> 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> 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> 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]]) |
- 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 |
- 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> 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> 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> 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 |
- 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> A = [1 2 3; 4 5 6; 7 8 9] M> A(1,1) ans = 1 |
- P> A = np.array([ [1,2,3], [4,5,6], [7,8,9] ]) P> A[0,0] 1 |
- R> A = matrix(c(1,2,3,4,5,9,7,8,9),nrow=3,byrow=T)
R> A[1,1] [1] 1 |
- 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 row to column vectors |
- M> b = [1 2 3]
M> b = b' b = 1 2 3 |
- 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> b = matrix(c(1,2,3), ncol=3) R> t(b) [,1] [1,] 1 [2,] 2 [3,] 3 |
- 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> 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 |
- 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]]) |
- 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 |
- 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> 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> 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]]) |
- 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> 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> 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 |
- 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]]) |
- 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> 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 = [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 |
- 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 |
- 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 |
- 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; |
- Matrix-scalar operations |
- ||||||
| Matrix-matrix multiplication |
- 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> 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> 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> 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> A = [1 2 3; 4 5 6; 7 8 9] M> b = [ 1; 2; 3 ] M> A * b ans = 14 32 50 |
- 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> 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> 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 = [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 |
- 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 |
- 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 |
- 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; |
- Element-wise matrix-matrix operations |
- ||||||
| Matrix elements to power n (here: individual elements squared) |
- 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> 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> 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> 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 = [1 2 3; 4 5 6; 7 8 9] M> A ^ 2 ans = 30 36 42 66 81 96 102 126 150 |
- 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> 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 |
- 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 = [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.T array([[1, 4, 7], [2, 5, 8], [3, 6, 9]]) |
- 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> 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> 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 = 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> 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> 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> 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> 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]]) |
- 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> 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> 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> cov( [x1,x2,x3] ) ans = 2.5000e-02 7.5000e-03 1.7500e-03 7.5000e-03 7.0000e-03 1.3500e-03 1.7500e-03 1.3500e-03 4.3000e-04 |
- 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> np.cov([x1, x2, x3]) Array([[ 0.025 , 0.0075 , 0.00175], [ 0.0075 , 0.007 , 0.00135], [ 0.00175, 0.00135, 0.00043]]) |
- 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> cov(matrix(c(x1, x2, x3), ncol=3)) [,1] [,2] [,3] [1,] 0.02500 0.00750 0.00175 [2,] 0.00750 0.00700 0.00135 [3,] 0.00175 0.00135 0.00043 |
- 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> A = [3 1; 1 3] A = 3 1 1 3 M> [eig_vec,eig_val] = eig(A) eig_vec = -0.70711 0.70711 0.70711 0.70711 eig_val = Diagonal Matrix 2 0 0 4 |
- 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> eig_val array([ 4., 2.]) P> eig_vec Array([[ 0.70710678, -0.70710678], [ 0.70710678, 0.70710678]]) |
- 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> 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 % ~/Desktop/io-2.0.2.tar.gz % pkg install % ~/Desktop/statistics-1.2.3.tar.gz M> pkg load statistics M> mean = [0 0] M> cov = [2 0; 0 2] cov = 2 0 0 2 M> mvnrnd(mean,cov,5) 2.480150 -0.559906 -2.933047 0.560212 0.098206 3.055316 -0.985215 -0.990936 1.122528 0.686977 |
- P> mean = np.array([0,0]) P> cov = np.array([[2,0],[0,2]]) P> np.random.multivariate_normal(mean, cov, 5) Array([[ 1.55432624, -1.17972629], [-2.01185294, 1.96081908], [-2.11810813, 1.45784216], [-2.93207591, -0.07369322], [-1.37031244, -1.18408792]]) |
- # requires the ‘mass’ package R> install.packages('MASS') R> library(MASS) R> mvrnorm(n=10, mean, cov) [,1] [,2] [1,] -0.8407830 -0.1882706 [2,] 0.8496822 -0.7889329 [3,] -0.1564171 0.8422177 [4,] -0.6288779 1.0618688 [5,] -0.5103879 0.1303697 [6,] 0.8413189 -0.1623758 [7,] -1.0495466 -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> mean=[0., 0.] 2-element Array{Float64,1}: 0.0 0.0 J> cov=[2. 0.; 0. 2.] 2x2 Array{Float64,2}: 2.0 0.0 0.0 2.0 J> rand( MvNormal(mean, cov), 5) 2x5 Array{Float64,2}: -0.527634 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 matrix (here: 5 random vectors with mean 0, covariance = 0, variance = 2) |
+ ||||||
|
+ Generating
+ a Gaussian dataset: |
+
+ %
+ requires statistics toolbox package |
+
+ P>
+ mean = np.array([0,0]) |
+
+ #
+ requires the ‘mass’ package |
+
+ #
+ requires the Distributions package from
+ https://github.com/JuliaStats/Distributions.jl |
+
+ Generating
+ a Gaussian dataset: |
||||||
* " operator would perform a matrix-matrix multiplication of NumPy matrices - same operator performs element-wise multiplication on NumPy arrays.
Vice versa, the "`.dot()`" method is used for element-wise multiplication of NumPy matrices, wheras the equivalent operation would for NumPy arrays would be achieved via the " * "-operator.
**Most people recommend the usage of the NumPy array type over NumPy matrices, since arrays are what most of the NumPy functions return.**
\ No newline at end of file
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+
+
+
+
+ |
+ Task + |
+
+ MATLAB/Octave + |
+
+ Python + NumPy + |
+
+ R + |
+
+ Julia + |
+
+ Task + |
+
|
+ CREATING + MATRICES + |
+ |||||
|
+ Creating
+ Matrices |
+
+ M>
+ A = [1 2 3; 4 5 6; 7 8 9] |
+
+ P>
+ A = np.array([ [1,2,3], [4,5,6], [7,8,9] ]) |
+
+ R>
+ A = matrix(c(1,2,3,4,5,6,7,8,9),nrow=3,byrow=T) |
+
+ J>
+ A=[1 2 3; 4 5 6; 7 8 9] |
+
+ Creating
+ Matrices |
+
|
+ Creating + an 1D column vector + |
+
+ M>
+ a = [1; 2; 3] |
+
+ P>
+ a = np.array([[1],[2],[3]]) |
+
+ R>
+ a = matrix(c(1,2,3), nrow=3, byrow=T) |
+
+ J>
+ a=[1; 2; 3] |
+
+ Creating + an 1D column vector + |
+
|
+ Creating
+ an |
+
+ M>
+ b = [1 2 3] |
+
+ P>
+ b = np.array([1,2,3]) |
+
+ R>
+ b = matrix(c(1,2,3), ncol=3) |
+
+ J>
+ b=[1 2 3] |
+
+ Creating
+ an |
+
|
+ Creating
+ a |
+
+ M>
+ rand(3,2) |
+
+ P>
+ np.random.rand(3,2) |
+
+ R>
+ matrix(runif(3*2), ncol=2) |
+
+ J>
+ rand(3,2) |
+
+ Creating
+ a |
+
|
+ Creating
+ a |
+
+ M>
+ zeros(3,2) |
+
+ P>
+ np.zeros((3,2)) |
+
+ R>
+ mat.or.vec(3, 2) |
+
+ J>
+ zeros(3,2) |
+
+ Creating
+ a |
+
|
+ Creating
+ an |
+
+ M>
+ ones(3,2) |
+
+ P>
+ np.ones([3,2]) |
+
+ R>
+ mat.or.vec(3, 2) + 1 |
+
+ J>
+ ones(3,2) |
+
+ Creating
+ an |
+
|
+ Creating
+ an |
+
+ M>
+ eye(3) |
+
+ P>
+ np.eye(3) |
+
+ R>
+ diag(3) |
+
+ J>
+ eye(3) |
+
+ Creating
+ an |
+
|
+ Creating
+ a |
+
+ M>
+ a = [1 2 3] |
+
+ P>
+ a = np.array([1,2,3]) |
+
+ R>
+ diag(1:3) |
+
+ J>
+ a=[1, 2, 3] |
+
+ Creating
+ a |
+
|
+ ACCESSING + MATRIX ELEMENTS + |
+ |||||
|
+ Getting
+ the dimension |
+
+ M>
+ A = [1 2 3; 4 5 6] |
+
+ P>
+ A = np.array([ [1,2,3], [4,5,6] ]) |
+
+ R>
+ A = matrix(1:6,nrow=2,byrow=T) R>
+ dim(A) |
+
+ J>
+ A=[1 2 3; 4 5 6] |
+
+ Getting
+ the dimension |
+
|
+ Selecting + rows + |
+
+ M>
+ A = [1 2 3; 4 5 6; 7 8 9] |
+
+ P>
+ A = np.array([ [1,2,3], [4,5,6], [7,8,9] ]) |
+
+ R>
+ A = matrix(1:9,nrow=3,byrow=T)
|
+
+ J>
+ A=[1 2 3; 4 5 6; 7 8 9]; |
+
+ Selecting + rows + |
+
|
+ Selecting + columns + |
+
+ M>
+ A = [1 2 3; 4 5 6; 7 8 9] |
+
+ P>
+ A = np.array([ [1,2,3], [4,5,6], [7,8,9] ]) |
+
+ R>
+ A = matrix(1:9,nrow=3,byrow=T)
|
+
+ J>
+ A=[1 2 3; 4 5 6; 7 8 9]; |
+
+ Selecting + columns + |
+
|
+ Extracting
+ rows and columns by criteria |
+
+ M>
+ A = [1 2 3; 4 5 9; 7 8 9] |
+
+ P>
+ A = np.array([ [1,2,3], [4,5,9], [7,8,9]]) |
+
+ R>
+ A = matrix(1:9,nrow=3,byrow=T)
|
+
+ J>
+ A=[1 2 3; 4 5 9; 7 8 9] |
+
+ Extracting
+ rows and columns by criteria |
+
|
+ Accessing
+ elements |
+
+ M>
+ A = [1 2 3; 4 5 6; 7 8 9] |
+
+ P>
+ A = np.array([ [1,2,3], [4,5,6], [7,8,9] ]) |
+
+ R>
+ A = matrix(c(1,2,3,4,5,9,7,8,9),nrow=3,byrow=T)
|
+
+ J>
+ A=[1 2 3; 4 5 6; 7 8 9]; |
+
+ Accessing
+ elements |
+
|
+ MANIPULATING + SHAPE AND DIMENSIONS + |
+ |||||
|
+ Converting |
+
+ M>
+ b = [1 2 3]
|
+
+ P>
+ b = np.array([1, 2, 3]) |
+
+ R>
+ b = matrix(c(1,2,3), ncol=3) |
+
+ J>
+ b=vec([1 2 3]) |
+
+ Converting |
+
|
+ Reshaping
+ Matrices |
+
+ M>
+ A = [1 2 3; 4 5 6; 7 8 9] |
+
+ P>
+ A = np.array([[1,2,3],[4,5,6],[7,8,9]]) |
+
+ R>
+ A = matrix(1:9,nrow=3,byrow=T)
|
+
+ J>
+ A=[1 2 3; 4 5 6; 7 8 9] |
+
+ Reshaping
+ Matrices |
+
|
+ Concatenating + matrices + |
+
+ M>
+ A = [1 2 3; 4 5 6] |
+
+ P>
+ A = np.array([[1, 2, 3], [4, 5, 6]]) |
+
+ R>
+ A = matrix(1:6,nrow=2,byrow=T) |
+
+ J>
+ A=[1 2 3; 4 5 6]; |
+
+ Concatenating + matrices + |
+
|
+ Stacking |
+
+ M>
+ a = [1 2 3] |
+
+ P>
+ a = np.array([1,2,3]) |
+
+ R>
+ a = matrix(1:3, ncol=3) |
+
+ J>
+ a=[1 2 3]; |
+
+ Stacking |
+
|
+ BASIC + MATRIX OPERATIONS + |
+ |||||
|
+ Matrix-scalar |
+
+ M> A
+ = [1 2 3; 4 5 6; 7 8 9] |
+
+ P>
+ A = np.array([ [1,2,3], [4,5,6], [7,8,9] ]) |
+
+ R>
+ A = matrix(1:9, nrow=3, byrow=T) R>
+ A + 2 |
+
+ J>
+ A=[1 2 3; 4 5 6; 7 8 9]; |
+
+ Matrix-scalar |
+
|
+ Matrix-matrix |
+
+ M> A
+ = [1 2 3; 4 5 6; 7 8 9] |
+
+ P>
+ A = np.array([ [1,2,3], [4,5,6], [7,8,9] ]) |
+
+ R>
+ A = matrix(1:9, nrow=3, byrow=T) |
+
+ J>
+ A=[1 2 3; 4 5 6; 7 8 9]; |
+
+ Matrix-matrix |
+
|
+ Matrix-vector |
+
+ M>
+ A = [1 2 3; 4 5 6; 7 8 9] |
+
+ P>
+ A = np.array([ [1,2,3], [4,5,6], [7,8,9] ]) |
+
+ R>
+ A = matrix(1:9, ncol=3) |
+
+ J>
+ A=[1 2 3; 4 5 6; 7 8 9]; |
+
+ Matrix-vector |
+
|
+ Element-wise |
+
+ M> A
+ = [1 2 3; 4 5 6; 7 8 9] |
+
+ P>
+ A = np.array([ [1,2,3], [4,5,6], [7,8,9] ]) |
+
+ R>
+ A = matrix(1:9, nrow=3, byrow=T)
|
+
+ J>
+ A=[1 2 3; 4 5 6; 7 8 9]; |
+
+ Element-wise |
+
|
+ Matrix
+ elements to power n |
+
+ M> A
+ = [1 2 3; 4 5 6; 7 8 9] |
+
+ P>
+ A = np.array([ [1,2,3], [4,5,6], [7,8,9] ]) |
+
+ R>
+ A = matrix(1:9, nrow=3, byrow=T) |
+
+ J>
+ A=[1 2 3; 4 5 6; 7 8 9]; |
+
+ Matrix
+ elements to power n |
+
|
+ Matrix
+ to power n |
+
+ M> A
+ = [1 2 3; 4 5 6; 7 8 9] |
+
+ P>
+ A = np.array([ [1,2,3], [4,5,6], [7,8,9] ]) |
+
+ R>
+ A = matrix(1:9, ncol=3) |
+
+ J>
+ A=[1 2 3; 4 5 6; 7 8 9]; |
+
+ Matrix
+ to power n |
+
|
+ Matrix + transpose + |
+
+ M> A
+ = [1 2 3; 4 5 6; 7 8 9] |
+
+ P>
+ A = np.array([ [1,2,3], [4,5,6], [7,8,9] ]) |
+
+ R>
+ A = matrix(1:9, nrow=3, byrow=T)
|
+
+ J>
+ A=[1 2 3; 4 5 6; 7 8 9] |
+
+ Matrix + transpose + |
+
|
+ Determinant
+ of a matrix: |
+
+ M>
+ A = [6 1 1; 4 -2 5; 2 8 7] |
+
+ P> A
+ = np.array([[6,1,1],[4,-2,5],[2,8,7]]) |
+
+ R>
+ A = matrix(c(6,1,1,4,-2,5,2,8,7), nrow=3, byrow=T) |
+
+ J>
+ A=[6 1 1; 4 -2 5; 2 8 7] |
+
+ Determinant
+ of a matrix: |
+
|
+ Inverse + of a matrix + |
+
+ M>
+ A = [4 7; 2 6] |
+
+ P>
+ A = np.array([[4, 7], [2, 6]]) |
+
+ R>
+ A = matrix(c(4,7,2,6), nrow=2, byrow=T) |
+
+ J>
+ A=[4 7; 2 6] |
+
+ Inverse + of a matrix + |
+
|
+ ADVANCED + MATRIX OPERATIONS + |
+ |||||
|
+ Calculating
+ the covariance matrix |
+
+ M>
+ x1 = [4.0000 4.2000 3.9000 4.3000 4.1000]’ |
+
+ P>
+ x1 = np.array([ 4, 4.2, 3.9, 4.3, 4.1]) |
+
+ R>
+ x1 = matrix(c(4, 4.2, 3.9, 4.3, 4.1), ncol=5) |
+
+ J>
+ x1=[4.0 4.2 3.9 4.3 4.1]'; |
+
+ Calculating
+ the covariance matrix |
+
|
+ Calculating |
+
+ M>
+ A = [3 1; 1 3] |
+
+ P>
+ A = np.array([[3, 1], [1, 3]]) |
+
+ R>
+ A = matrix(c(3,1,1,3), ncol=2) |
+
+ J>
+ A=[3 1; 1 3] |
+
+ Calculating |
+
|
+ Generating
+ a Gaussian dataset: |
+
+ %
+ requires statistics toolbox package |
+
+ P>
+ mean = np.array([0,0]) |
+
+ #
+ requires the ‘mass’ package |
+
+ #
+ requires the Distributions package from
+ https://github.com/JuliaStats/Distributions.jl |
+
+ Generating
+ a Gaussian dataset: |
+
+