numpyには（他のライブラリにも）QR分解あるけどQL分解がないので簡単に作る．

行列生成

import numpy as np

A = np.random.rand(4,4)
print(A)

output

[[ 0.79124883  0.99846239  0.68643175  0.62022478]
 [ 0.62550988  0.4907801   0.43422796  0.70191408]
 [ 0.55936063  0.31190068  0.77154881  0.08427772]
 [ 0.11253189  0.65420468  0.46686985  0.11447206]]

QR分解

QR分解

Q, R = np.linalg.qr(A)
print('Q\n', Q)
print('R\n', R)
print('Q @ R\n', Q @ R)
print('A == Q @ R ?', np.allclose(A, Q @ R))
print('Q @ Q^T == I ?', np.allclose(Q @ Q.transpose(), np.eye(4)))
print('Q^T @ Q == I ?', np.allclose(Q.transpose() @ Q, np.eye(4)))

output

Q
 [[-0.68280023  0.31934702  0.36684631 -0.54518356]
 [-0.53977747 -0.21055993  0.30305429  0.75661278]
 [-0.48269464 -0.38522751 -0.77364703 -0.14168954]
 [-0.09710826  0.83981055 -0.41839235  0.33201818]]
R
 [[-1.15882918 -1.16074386 -1.12084163 -0.85416362]
 [ 0.          0.64477264  0.22263932  0.1139407 ]
 [ 0.          0.         -0.40883161  0.32714981]
 [ 0.          0.          0.          0.21900634]]
Q @ R
 [[ 0.79124883  0.99846239  0.68643175  0.62022478]
 [ 0.62550988  0.4907801   0.43422796  0.70191408]
 [ 0.55936063  0.31190068  0.77154881  0.08427772]
 [ 0.11253189  0.65420468  0.46686985  0.11447206]]
A == Q @ R ? True
Q @ Q^T == I ? True
Q^T @ Q == I ? True

QL分解

QL分解

def QL(A):
    """
    QL factorization by using QR factorization
    """
    Q, L = np.linalg.qr(A[::-1,::-1])
    return Q[::-1,::-1], L[::-1,::-1]


Q, L = QL(A)
print('Q\n', Q)
print('L\n', L)
print('Q @ L\n', Q @ L)
print('A == Q @ L ?', np.allclose(A, Q @ L))
print('Q @ Q^T == I ?', np.allclose(Q @ Q.transpose(), np.eye(4)))
print('Q^T @ Q == I ?', np.allclose(Q.transpose() @ Q, np.eye(4)))

output

Q
 [[ 0.61195303  0.42670519 -0.1218915  -0.65465917]
 [-0.44726347 -0.4150739   0.28064256 -0.74088379]
 [ 0.09753892 -0.51781405 -0.84524639 -0.08895675]
 [-0.64494576  0.61441431 -0.43811043 -0.12082746]]
L
 [[ 0.18642187  0.          0.          0.        ]
 [-0.14250644  0.46278524  0.          0.        ]
 [-0.44300076 -0.53421711 -0.81849675  0.        ]
 [-1.04478428 -1.12405514 -0.89613647 -0.94740105]]
Q @ L
 [[ 0.79124883  0.99846239  0.68643175  0.62022478]
 [ 0.62550988  0.4907801   0.43422796  0.70191408]
 [ 0.55936063  0.31190068  0.77154881  0.08427772]
 [ 0.11253189  0.65420468  0.46686985  0.11447206]]
A == Q @ L ? True
Q @ Q^T == I ? True
Q^T @ Q == I ? True