Linear Algebra

Contents

Linear Algebra#

Perron–Frobenius theorem#

Theorem 1 (Perron–Frobenius theorem)

a real square matrix with positive entries has a unique largest real eigenvalue and that the corresponding eigenvector can be chosen to have strictly positive components, and also asserts a similar statement for certain classes of nonnegative matrices.

Matrix dot product#

import numpy as np
from math import sqrt

U = np.matrix([[1/sqrt(2)+1j/2, -1j/2], 
              [1/2, 1/2+1j/sqrt(2)]])
X = np.matrix([[0,1],[1,0]])
Y = np.matrix([[0,-1j],[1j,0]])
Z = np.matrix([[1,0],[0,-1]])
# U.getH()
np.matmul(U,X)
print(np.matmul(U,X,U.H))
print(np.matmul(np.dot(U,X),U.getH()))
print(np.matmul(np.matmul(U,X),U.H))
print(np.dot(np.dot(U,X),U.getH()))
print(np.dot(np.dot(U,Y),U.getH()))
print(np.dot(np.dot(U,Z),U.getH()))

[[0.        -0.5j        0.70710678+0.5j       ]
 [0.5       +0.70710678j 0.5       +0.j        ]]
[[-0.5       +0.j   0.70710678-0.5j]
 [ 0.70710678+0.5j  0.5       +0.j ]]
[[-0.5       +0.j   0.70710678-0.5j]
 [ 0.70710678+0.5j  0.5       +0.j ]]
[[-0.5       +0.j   0.70710678-0.5j]
 [ 0.70710678+0.5j  0.5       +0.j ]]
[[ 7.07106781e-01+0.j          8.86511593e-17-0.70710678j]
 [ 1.11022302e-16+0.70710678j -7.07106781e-01+0.j        ]]
[[ 0.5       +0.j   0.70710678+0.5j]
 [ 0.70710678-0.5j -0.5       +0.j ]]

print(np.matmul(U,U.getH()))

[[1.+0.j 0.+0.j]
 [0.+0.j 1.+0.j]]

print(np.dot(U,U.getH()))

[[1.+0.j 0.+0.j]
 [0.+0.j 1.+0.j]]

Matrix exponentiation#

from scipy.linalg import expm, sinm, cosm
import numpy as np
# hadmard gate
h = np.array([[1.0, 1.0], [1.0, -1.0]])
expm(1j*h/np.sqrt(2)*np.pi/2)

array([[6.123234e-17+0.70710678j, 0.000000e+00+0.70710678j],
       [0.000000e+00+0.70710678j, 6.123234e-17-0.70710678j]])

Solve matrix inverse#

## solve A(4,3)*B(3,2)=C(4,2)
import numpy as np
from numpy.linalg import inv, matrix_rank, det
from numpy import dot, transpose
# np.set_printoptions(precision=5)

A = np.random.randn(4, 3)
C = np.random.randn(4, 2)
print(matrix_rank(A))
print('A=',A)
print('C=',C)
AT_A = dot(transpose(A),A)
print('AT_A=',AT_A)
print('rank of AT_A:',matrix_rank(AT_A))
A_left_inverse = dot(inv( AT_A ), transpose(A))
print(dot(inv( AT_A ), AT_A ))
print(dot(A,A_left_inverse))
B = dot(A_left_inverse, C)
print('B=',B)
# print(dot(A,dot(A_left_inverse, C)))
print('A*B=',dot(A,B))

3
A= [[-0.58467778  0.82158951  0.10574808]
 [-0.50681406  0.4223986   0.42152069]
 [-0.04623778  0.53622639 -0.99689905]
 [ 0.44557443  0.77302169 -0.43747922]]
C= [[-0.7486619  -0.82060587]
 [-0.72422963  0.28541297]
 [-1.23958652  0.95226842]
 [ 0.66498058 -0.7678366 ]]
AT_A= [[ 0.7993831  -0.3747979  -0.42429632]
 [-0.3747979   1.73853117 -0.60781324]
 [-0.42429632 -0.60781324  1.37405814]]
rank of AT_A: 3
[[ 1.00000000e+00  1.60143113e-16 -2.82515649e-16]
 [ 7.02571532e-17  1.00000000e+00 -1.73973845e-16]
 [ 1.17481372e-16  7.45963956e-18  1.00000000e+00]]
[[ 0.62772876  0.46182522  0.13355035  0.05066688]
 [ 0.46182522  0.42707759 -0.16567737 -0.06285536]
 [ 0.13355035 -0.16567737  0.95208952 -0.01817648]
 [ 0.05066688 -0.06285536 -0.01817648  0.99310413]]
B= [[ 2.24272868 -1.26220417]
 [ 0.29783903 -1.09075364]
 [ 1.232111   -1.29426549]]
A*B= [[-0.93627883 -0.2950351 ]
 [-0.49147935 -0.36658968]
 [-1.17227993  0.76372268]
 [ 0.69051563 -0.83936786]]

a = np.array([[1., 2., 3.], [3., 5., 3.]])
aT_a = dot(transpose(a),a)
print(aT_a)
print(det(aT_a))
print('rank:',matrix_rank(aT_a))
ainv = inv(aT_a)
print(ainv)
np.allclose(dot(ainv, aT_a), np.eye(3))

[[10. 17. 12.]
 [17. 29. 21.]
 [12. 21. 18.]]
1.6710489201527266e-14
rank: 2
[[ 4.84725486e+15 -3.23150324e+15  5.38583873e+14]
 [-3.23150324e+15  2.15433549e+15 -3.59055916e+14]
 [ 5.38583873e+14 -3.59055916e+14  5.98426526e+13]]

False