#### Qr Decomposition

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```import numpy as np

def qr_householder(A):
"""Return a QR-decomposition of the matrix A using Householder reflection.

The QR-decomposition decomposes the matrix A of shape (m, n) into an
orthogonal matrix Q of shape (m, m) and an upper triangular matrix R of
shape (m, n).  Note that the matrix A does not have to be square.  This
method of decomposing A uses the Householder reflection, which is
numerically stable and of complexity O(n^3).

https://en.wikipedia.org/wiki/QR_decomposition#Using_Householder_reflections

Arguments:
A -- a numpy.ndarray of shape (m, n)

Note: several optimizations can be made for numeric efficiency, but this is
intended to demonstrate how it would be represented in a mathematics
textbook.  In cases where efficiency is particularly important, an optimized
version from BLAS should be used.

>>> A = np.array([[12, -51, 4], [6, 167, -68], [-4, 24, -41]], dtype=float)
>>> Q, R = qr_householder(A)

>>> # check that the decomposition is correct
>>> np.allclose(Q@R, A)
True

>>> # check that Q is orthogonal
>>> np.allclose(Q@Q.T, np.eye(A.shape))
True
>>> np.allclose(Q.T@Q, np.eye(A.shape))
True

>>> # check that R is upper triangular
>>> np.allclose(np.triu(R), R)
True
"""
m, n = A.shape
t = min(m, n)
Q = np.eye(m)
R = A.copy()

for k in range(t - 1):
# select a column of modified matrix A':
x = R[k:, [k]]
# construct first basis vector
e1 = np.zeros_like(x)
e1 = 1.0
# determine scaling factor
alpha = np.linalg.norm(x)
# construct vector v for Householder reflection
v = x + np.sign(x) * alpha * e1
v /= np.linalg.norm(v)

# construct the Householder matrix
Q_k = np.eye(m - k) - 2.0 * v @ v.T
# pad with ones and zeros as necessary
Q_k = np.block([[np.eye(k), np.zeros((k, m - k))], [np.zeros((m - k, k)), Q_k]])

Q = Q @ Q_k.T
R = Q_k @ R

return Q, R

if __name__ == "__main__":
import doctest

doctest.testmod()
```  