"""Return a QR-decomposition of the matrix A using Householder reflection.

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[0] = 1.0

# determine scaling factor

alpha = np.linalg.norm(x)

# construct vector v for Householder reflection

v = x + np.sign(x[0]) * 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