linalg.qr

Description

Compute the QR factorization of a matrix.

Factor the matrix a as q and r matrices, where q is orthonormal and r is upper-triangular.

Mandatory Input Parameters

Parameter	Type	Description
a	array_like, shape (…, M, N)	An array-like object with the dimensionality of at least 2

Optional Input Parameters

Parameter	Type	Default Value	Description
mode	{'reduced', 'complete', 'r', 'raw'}	reduced	If K = min(M, N), then: "reduced" returns q and r with dimensions (…, M, K) and (…, K, N) respectively. "complete" returns q and r with dimensions (...,M,M) and (...,M,N) respectively. "r" returns r only with dimensions (…, K, N). "raw" returns h and tau with dimensions (…, N, M) and (…, K,) respectively.

Parameter

Type

Default Value

Description

mode

{'reduced', 'complete', 'r', 'raw'}

reduced

If K = min(M, N), then:

"reduced" returns q and r with dimensions (…, M, K) and (…, K, N) respectively.
"complete" returns q and r with dimensions (...,M,M) and (...,M,N) respectively.
"r" returns r only with dimensions (…, K, N).
"raw" returns h and tau with dimensions (…, N, M) and (…, K,) respectively.

Return Value

Parameter	Type	Description
q	ndarray of float or complex	A matrix with orthonormal columns. When mode = "complete", the result is an orthogonal/unitary matrix depending on whether or not a is real/complex. The determinant may be either +/- 1 in that case. In case the number of dimensions in the input array is greater than 2, a stack of the matrices with above properties is returned.
r	ndarray of float or complex	An upper-triangular matrix or a stack of upper-triangular matrices if the number of dimensions in the input array is greater than 2.
(h, tau)	ndarrays of np.double or np.cdouble	The array h contains the Householder reflectors that generate q along with r. The tau array contains scaling factors for the reflectors.

Examples

>>> import numpy as np
>>> a = np.random.randn(9, 6)
>>> q, r = np.linalg.qr(a)
>>> 
>>> np.allclose(a, np.dot(q,r))
True
>>> 
>>> r2 = np.linalg.qr(a, mode='r')
>>> np.allclose(r, r2)
True
>>> 
>>> a = np.random.normal(size=(3,2,2))
>>> q, r = np.linalg.qr(a)
>>> q.shape
(3, 2, 2)
>>> r.shape
(3, 2, 2)
>>> np.allclose(a, np.matmul(q,r))
True
>>> 
>>> # Ax = b. If A = qr, then x = inv(r) * (q.T) * b.
>>> 
>>> A = np.array([[0,1], [1,1], [1,1], [2,1]])
>>> A
array([[0, 1],
       [1, 1],
       [1, 1],
       [2, 1]])
>>> b = np.array([1, 0, 2, 1])
>>> q, r = np.linalg.qr(A)
>>> p = np.dot(q.T, b)
>>> 
>>> np.dot(np.linalg.inv(r), p)
array([2.62287285e-16, 1.00000000e+00])
>>>

Parent topic: Linear Algebra Functions