Calculus: Compute a QR Factorization using the Householder Method

Date: 2013-11-11 |

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

A =

143
221
2-41
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 =

1/32/32/3
2/31/3-2/3
2/3-2/31/3
**Solution: A = QR**

**R = M1A = **

307/3
065/3
005/3
**Q = M1^t =**
1/32/32/3
2/31/3-2/3
2/3-2/31/3

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