from __future__ import division
import numpy as np
import numpy.linalg as la
n = 3
e1 = np.array([1,0,0])
e2 = np.array([0,1,0])
e3 = np.array([0,0,1])
A = np.random.randn(3, 3)
A
Householder reflector: \[I-2\frac{vv^T}{v^Tv}\]
Choose \(v=a-\|a\|e_1\).
a = A[:, 0]
v = a-la.norm(a)*e1
H1 = ...
A1 = np.dot(H1, A)
A1
(Edit this cell for solution.)
NB: Never build full Householder matrices in actual code! (Why? How?)
a = A1[:, 1].copy()
a[0] = 0
v = a-la.norm(a)*e2
H2 = ...
R = np.dot(H2, A1)
R
(Edit this cell for solution.)
Q = np.dot(H2, H1).T
la.norm(np.dot(Q, R) - A)