more FACINATING tidbits and DETAILS about

1. more FACINATING tidbitsand DETAILS about Householder TRANSFORMATIONS!

2. Householder Reflectors A Householder reflector is a matrix of the form

3. Q is nifty because it is both symmetric and orthogonal

4. Householder Matrix Householder matrices reduce zk+1 ,…,zn (sub-diagonals) to zero

5. To achieve this, v must be a linear combination of x and ek

6. The Transformation Part(take a deep breath)

7. And then..and then..(exciting isn't it?) One last thing here.. sign(above) just means you make it postitive or negative depending on what you need.

8. Householder matrix Corollary (kth Householder matrix):Let A be an nxn matrix and x any vector. If k is an integer with 1< k<n-1 we can construct a vector w(k) and matrix H(k) = I - 2w(k)w’(k) so that

9. Define the value  so that

10. Furthermore….

11. So…putting this all together

12. An exceptionally tedious example, well at least the first transform of one…(please note A is NOT symmetric)