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# scipy.linalg.qr

`scipy.linalg.``qr`(a, overwrite_a=False, lwork=None, mode='full')

  Compute QR decomposition of a matrix.

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

  Parameters : | a : array, shape (M, N) Matrix to be decomposed overwrite_a : bool, optional Whether data in a is overwritten (may improve performance) lwork : int, optional Work array size, lwork >= a.shape[1]. If None or -1, an optimal size is computed. mode : {‘full’, ‘r’, ‘economic’} 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).
  Returns : | Q : double or complex ndarray Of shape (M, M), or (M, K) for `mode='economic'`. Not returned if `mode='r'`. R : double or complex ndarray Of shape (M, N), or (K, N) for `mode='economic'`. `K = min(M, N)`. Raises LinAlgError if decomposition fails :

  Notes

  This is an interface to the LAPACK routines dgeqrf, zgeqrf, dorgqr, and zungqr.

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

  Examples

      >>> from scipy import random, linalg, dot
      >>> a = random.randn(9, 6)
      >>> q, r = linalg.qr(a)
      >>> allclose(a, dot(q, r))
      True
      >>> q.shape, r.shape
      ((9, 9), (9, 6))

      >>> r2 = linalg.qr(a, mode='r')
      >>> allclose(r, r2)

      >>> q3, r3 = linalg.qr(a, mode='economic')
      >>> q3.shape, r3.shape
      ((9, 6), (6, 6))

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