scipy.linalg.rq(a, overwrite_a=False, lwork=None, mode='full', check_finite=True)[source]#

Compute RQ decomposition of a matrix.

Calculate the decomposition A = R Q where Q is unitary/orthogonal and R upper triangular.

a(M, N) array_like

Matrix to be decomposed

overwrite_abool, optional

Whether data in a is overwritten (may improve performance)

lworkint, optional

Work array size, lwork >= a.shape[1]. If None or -1, an optimal size is computed.

mode{‘full’, ‘r’, ‘economic’}, optional

Determines what information is to be returned: either both Q and R (‘full’, default), only R (‘r’) or both Q and R but computed in economy-size (‘economic’, see Notes).

check_finitebool, optional

Whether to check that the input matrix contains only finite numbers. Disabling may give a performance gain, but may result in problems (crashes, non-termination) if the inputs do contain infinities or NaNs.

Rfloat or complex ndarray

Of shape (M, N) or (M, K) for mode='economic'. K = min(M, N).

Qfloat or complex ndarray

Of shape (N, N) or (K, N) for mode='economic'. Not returned if mode='r'.


If decomposition fails.


This is an interface to the LAPACK routines sgerqf, dgerqf, cgerqf, zgerqf, sorgrq, dorgrq, cungrq and zungrq.

If mode=economic, the shapes of Q and R are (K, N) and (M, K) instead of (N,N) and (M,N), with K=min(M,N).


>>> import numpy as np
>>> from scipy import linalg
>>> rng = np.random.default_rng()
>>> a = rng.standard_normal((6, 9))
>>> r, q = linalg.rq(a)
>>> np.allclose(a, r @ q)
>>> r.shape, q.shape
((6, 9), (9, 9))
>>> r2 = linalg.rq(a, mode='r')
>>> np.allclose(r, r2)
>>> r3, q3 = linalg.rq(a, mode='economic')
>>> r3.shape, q3.shape
((6, 6), (6, 9))