Talk:Householder transformation

applications

The applications given feel a bit out of place. Maybe it could be given in the page about tridiagonalization, and link to that here?— Preceding unsigned comment added by 157.92.44.71 (talk) 21:02, 18 October 2010 (UTC)[reply]

I'm guessing this was written before as many applications were added as they are now. Fephisto (talk) 03:38, 5 February 2025 (UTC)[reply]

orthogonalisation

If this is an alternative to the Gram-Schmidt process, perhaps this would be a good place for an example and brief discussion of relative merit? Cesiumfrog (talk) 22:40, 23 November 2010 (UTC)[reply]

The Householder transformation is mentioned on the Non-Linear Least Squares Method, as well as the QR decomposition page. There should be a link to these pages.. —Preceding unsigned comment added by 187.2.114.212 (talk) 13:27, 30 April 2011 (UTC)[reply]

There is now a QR decomposition section, which does link over to those pages now. Fephisto (talk) 01:44, 4 February 2025 (UTC)[reply]

Properties

This part is VERY poorly explained: "A Householder matrix has eigenvalues \pm 1. To see this, notice that if u is orthogonal to the vector v which was used to create the reflector, then Pu = u, i.e., 1 is an eigenvalue of multiplicity n − 1, since there are n − 1 vectors orthogonal to u. Also, notice Pv = − v, and so -1 is an eigenvalue with multiplicity 1." Nowhere does the writer explain why Pu=u. —Preceding unsigned comment added by 173.248.196.243 (talk) 18:58, 15 February 2011 (UTC)[reply]

Well, 14 years later, I hope it's a bit better now...Fephisto (talk) 18:54, 15 January 2025 (UTC)[reply]

Example section is useless

I'm currently looking for *any* example of an actual Householder transformation. Sadly not even the Example section on the Wikipedia article of the Householder transformation shows the actual steps of the Householder transformation. The whole section is useless and may very well be removed. 37.120.17.148 (talk) 15:40, 17 November 2015 (UTC)[reply]

A decade later, finally added. Better late than never :/. Fephisto (talk) 03:37, 5 February 2025 (UTC)[reply]

Use of inner product

Can one take the inner product of a point and a vector, $<x,v>$ ? This is written when finding the reflection of $x$ . — Preceding unsigned comment added by Healyp (talk • contribs) 11:16, 31 March 2017 (UTC)[reply]

Where in the article is this? Fephisto (talk) 21:36, 3 February 2025 (UTC)[reply]

Oh, in the VERY beginning. I see what's going on now. That's valid. No, it's not. It's doing a shorthand where they're considering the vector 'x' to be the vector from the origin to the point x. I'll try to work through clarifying this. Fephisto (talk) 03:37, 5 February 2025 (UTC)[reply]

This should hopefully be done now. Fephisto (talk) 03:49, 5 February 2025 (UTC)[reply]

The part about tridiagonalization

This is actually about reduction to Hessenberg form, the result of which is tridiagonal if the input matrix A is symmetric. While the process uses Householder reflectors, it uses it in a slightly modified form. There should be a general example about Householder reflectors, not such a specific use-case. Tlkdrt (talk) 21:28, 28 October 2022 (UTC)[reply]

Correct. This has been rectified. Fephisto (talk) 21:36, 3 February 2025 (UTC)[reply]