from numpy import inf
from numpy import array
from numpy import identity
from numpy import trace
from numpy import count_nonzero
from numpy import tensordot
from numpy import tril, triu, diag
from numpy.linalg import norm, inv, det, matrix_rank
from scipy.sparse import csr_matrixMatrices
Vectors and Vector Arithmetic
Defining a Vector
vector = array([1,2,3])
print(vector)[1 2 3]Vector Arithmetic
Vector Addition
a = array([1,2,3])
print(a)[1 2 3]b = array([4,5,6])
print(b)[4 5 6]c = a + b
print(c)[5 7 9]Vector Subtraction
a = array([1,2,3])
print(a)[1 2 3]b = array([0.5,0.5,0.5])
print(b)[0.5 0.5 0.5]c = a - b
print(c)[0.5 1.5 2.5]Vector Multiplication
a = array([1,2,3])
print(a)[1 2 3]b = array([1,2,3])
print(b)[1 2 3]c = a*b
print(c)[1 4 9]Vector Division
a = array([1,2,3])
print(a)[1 2 3]b = array([1,2,3])
print(b)[1 2 3]c = a/b
print(c)[1. 1. 1.]Vector Dot Product
a = array([1,2,3])
print(a)[1 2 3]b = array([1,2,3])
print(b)[1 2 3]c = a.dot(b)
print(c)14Vector-Scalar Multiplication
a = array([1,2,3])
print(a)[1 2 3]b = 0.5
print(b)0.5c = b*a
print(c)[0.5 1. 1.5]Vector Norms
a = array([1,2,3])
print(a)[1 2 3]Vector L1 Norm
l1 = norm(a, 1)
print(l1)6.0Vector L2 Norm
l2 = norm(a)
print(l2)3.7416573867739413Vector Max Norm
maxnorm = norm(a,inf)
print(maxnorm)3.0Matrices and Matrix Arithmetic
Defining a Matrix
A = array([[1,2,3],
[4,5,6]])
print(A)[[1 2 3]
[4 5 6]]Matrix Arithmetic
A = array([[1,2,3],
[4,5,6]])
print(A)[[1 2 3]
[4 5 6]]B = array([[1,2,3],
[4,5,6]])
print(B)[[1 2 3]
[4 5 6]]Matrix Addition
C = A + B
print(C)[[ 2 4 6]
[ 8 10 12]]Matrix Subtraction
C = A - B
print(C)[[0 0 0]
[0 0 0]]Matrix Multiplication (Hadamard Product)
C = A *B
print(C)[[ 1 4 9]
[16 25 36]]Matrix Division
C = A / B
print(C)[[1. 1. 1.]
[1. 1. 1.]]Matrix-Matrix Multiplication
A = array([
[1,2],
[3,4],
[5,6]])
print(A)[[1 2]
[3 4]
[5 6]]B = array([
[1,2],
[3,4]])
print(B)[[1 2]
[3 4]]C = A.dot(B)
print(C)[[ 7 10]
[15 22]
[23 34]]D = A @ B
print(D)[[ 7 10]
[15 22]
[23 34]]Matrix-Vector Multiplication
A = array([
[1,2],
[3,4],
[5,6]])
print(A)[[1 2]
[3 4]
[5 6]]b = array([0.5, 0.5])
print(b)[0.5 0.5]C = A.dot(b)
print(C)[1.5 3.5 5.5]Matrix-Scalar Multiplication
A = array([
[1,2],
[3,4],
[5,6]])
print(A)[[1 2]
[3 4]
[5 6]]b = 0.5
print(b)0.5C = A*b
print(C)[[0.5 1. ]
[1.5 2. ]
[2.5 3. ]]Types of Matrices
Square Matrix
Symmetric Matrix
Triangular Matrix
M = array([
[1,2,3],
[1,2,3],
[1,2,3]])
print(M)[[1 2 3]
[1 2 3]
[1 2 3]]lower = tril(M)
print(lower)[[1 0 0]
[1 2 0]
[1 2 3]]upper = triu(M)
print(upper)[[1 2 3]
[0 2 3]
[0 0 3]]Diagonal Matrix
M = array([
[1,2,3],
[1,2,3],
[1,2,3]])
print(M)[[1 2 3]
[1 2 3]
[1 2 3]]d = diag(M)
print(d)[1 2 3]D = diag(d)
print(D)[[1 0 0]
[0 2 0]
[0 0 3]]Identity Matrix
I = identity(3)
print(I)[[1. 0. 0.]
[0. 1. 0.]
[0. 0. 1.]]Orthogonal Matrix
Q = array([
[1,0],
[0,-1]])
print(Q)[[ 1 0]
[ 0 -1]]V = inv(Q)
print(V)[[ 1. 0.]
[-0. -1.]]print(Q.T)[[ 1 0]
[ 0 -1]]I = Q.dot(Q.T)
print(I)[[1 0]
[0 1]]Matrix Operations
Transpose
A = array([
[1,2],
[3,4],
[5,6]])
print(A)[[1 2]
[3 4]
[5 6]]C = A.T
print(C)[[1 3 5]
[2 4 6]]Inverse
A = array([
[1.0, 2.0],
[3.0, 4.0]])
print(A)[[1. 2.]
[3. 4.]]B = inv(A)
print(B)[[-2. 1. ]
[ 1.5 -0.5]]I = A.dot(B)
print(I)[[1.0000000e+00 0.0000000e+00]
[8.8817842e-16 1.0000000e+00]]Trace
A = array([
[1,2,3],
[4,5,6],
[7,8,9]])
print(A)[[1 2 3]
[4 5 6]
[7 8 9]]B = trace(A)
print(B)15Determinant
A = array([
[1,2,3],
[4,5,6],
[7,8,99]])
print(A)[[ 1 2 3]
[ 4 5 6]
[ 7 8 99]]B = det(A)
print(B)-269.99999999999983Rank
v1 = array([1,2,3])
print(v1)[1 2 3]print(matrix_rank(v1))1v2 = array([0,0,0,0,0])
print(v2)[0 0 0 0 0]print(matrix_rank(v2))0M0 = array([
[0,0],
[0,0]])
print(M0)[[0 0]
[0 0]]print(matrix_rank(M0))0M1 = array([
[1,2],
[1,2]])
print(M1)[[1 2]
[1 2]]print(matrix_rank(M1))1M2 = array([
[1,2],
[3,4]])
print(M2)[[1 2]
[3 4]]print(matrix_rank(M2))2Sparse Matrices
Sparse Matrices in Python
A = array([
[1,0,0,1,0,0],
[0,0,2,0,0,1],
[0,0,0,2,0,0]])
print(A)[[1 0 0 1 0 0]
[0 0 2 0 0 1]
[0 0 0 2 0 0]]S = csr_matrix(A)
print(S) (0, 0) 1
(0, 3) 1
(1, 2) 2
(1, 5) 1
(2, 3) 2B = S.todense()
print(B)[[1 0 0 1 0 0]
[0 0 2 0 0 1]
[0 0 0 2 0 0]]sparsity = 1.0 - count_nonzero(A) / A.size
print(sparsity)0.7222222222222222Tensors and Tensor Arithmetic
Tensors in Python
T = array([
[[1,2,3], [4,5,6], [7,8,9]],
[[11,12,13], [14,15,16], [17,18,19]],
[[21,22,23], [24,25,26], [27,28,29]]])
print(T)[[[ 1 2 3]
[ 4 5 6]
[ 7 8 9]]
[[11 12 13]
[14 15 16]
[17 18 19]]
[[21 22 23]
[24 25 26]
[27 28 29]]]print(T.shape)(3, 3, 3)Tensor Arithmetic
Tensor Addition
A = array([
[[1,2,3], [4,5,6], [7,8,9]],
[[11,12,13], [14,15,16], [17,18,19]],
[[21,22,23], [24,25,26], [27,28,29]]])
B = array([
[[1,2,3], [4,5,6], [7,8,9]],
[[11,12,13], [14,15,16], [17,18,19]],
[[21,22,23], [24,25,26], [27,28,29]]])
C = A + B
print(C)[[[ 2 4 6]
[ 8 10 12]
[14 16 18]]
[[22 24 26]
[28 30 32]
[34 36 38]]
[[42 44 46]
[48 50 52]
[54 56 58]]]Tensor Subtraction
A = array([
[[1,2,3], [4,5,6], [7,8,9]],
[[11,12,13], [14,15,16], [17,18,19]],
[[21,22,23], [24,25,26], [27,28,29]]])
B = array([
[[1,2,3], [4,5,6], [7,8,9]],
[[11,12,13], [14,15,16], [17,18,19]],
[[21,22,23], [24,25,26], [27,28,29]]])
C = A - B
print(C)[[[0 0 0]
[0 0 0]
[0 0 0]]
[[0 0 0]
[0 0 0]
[0 0 0]]
[[0 0 0]
[0 0 0]
[0 0 0]]]Tensor Hadamard Product
A = array([
[[1,2,3], [4,5,6], [7,8,9]],
[[11,12,13], [14,15,16], [17,18,19]],
[[21,22,23], [24,25,26], [27,28,29]]])
B = array([
[[1,2,3], [4,5,6], [7,8,9]],
[[11,12,13], [14,15,16], [17,18,19]],
[[21,22,23], [24,25,26], [27,28,29]]])
C = A * B
print(C)[[[ 1 4 9]
[ 16 25 36]
[ 49 64 81]]
[[121 144 169]
[196 225 256]
[289 324 361]]
[[441 484 529]
[576 625 676]
[729 784 841]]]Tensor Division
A = array([
[[1,2,3], [4,5,6], [7,8,9]],
[[11,12,13], [14,15,16], [17,18,19]],
[[21,22,23], [24,25,26], [27,28,29]]])
B = array([
[[1,2,3], [4,5,6], [7,8,9]],
[[11,12,13], [14,15,16], [17,18,19]],
[[21,22,23], [24,25,26], [27,28,29]]])
C = A / B
print(C)[[[1. 1. 1.]
[1. 1. 1.]
[1. 1. 1.]]
[[1. 1. 1.]
[1. 1. 1.]
[1. 1. 1.]]
[[1. 1. 1.]
[1. 1. 1.]
[1. 1. 1.]]]Tensor Product
A = array([1,2])
B = array([3,4])C = tensordot(A, B, axes=0)
print(C)[[3 4]
[6 8]]