Calculus: Compute a QR Factorization using the Householder Method

Compute a QR Factorization of A using the Householder Method.  Separately list all of the Householder reflections used along the way.

A =

1	4	3
2	2	1
2	-4	1

Start by finding the Householder reflection M with V1 = the first column of A

X = V1 = (1, 2, 2)

Y = ||X||e1 = (3, 0, 0)

U = (Y-X)/(||Y-X||) = (1/sqrt(3))(1, -1, -1)

M1 = I – 2 UU^t                                                           (^t denotes the matrix’s transpose)

M1 =

1/3	2/3	2/3
2/3	1/3	-2/3
2/3	-2/3	1/3

**Solution: A = QR**

**R = M1A = **

3	0	7/3
0	6	5/3
0	0	5/3

**Q = M1^t =**

1/3	2/3	2/3
2/3	1/3	-2/3
2/3	-2/3	1/3