Talk:Power iteration

From Wikipedia, the free encyclopedia

[edit] Finding the 2nd Eigenvalue

I would like to add the following - please let me know if there are any comments.

In order to find the second eigenvalue, the Power Method can be used on the following matrix:

A' = A - \lambda_1 w_1w_1^T

where

w_i = \frac{v_i}{|v_i|}.

The matrix A' has the same eigenvectors and eigenvalues as A with the exception that λ1 = 0. To see this we multiply by w1

\begin{align} A'w_1 &= Aw_1 - \lambda_1 w_1w_1^Tw_1\\ &= \lambda_1 w_1 - \lambda_1 w_1\\ &= 0 \end{align}

for any other eigenvector we have

\begin{align} A'w_2 &= Aw_2 - \lambda_1 w_1w_1^Tw_2\\ &= \lambda_1 w_1 \\ \end{align}

since w1 and w2 are orthogonal to each other. --Adieyal 15:05, 5 July 2007 (UTC)

Why are w1 and w2 orthogonal to each other? If I understand you correctly, w1 and w2 are (normalized) eigenvectors. Eigenvectors of a general matrix are not orthogonal, but eigenvalues of a symmetric matrix (and, more generally, a normal matrix) are. Perhaps you're assuming that A is symmetric? -- Jitse Niesen (talk) 20:48, 6 July 2007 (UTC)

Clearly you're right - I'm not sure why I thought that the matrix was symmetric. Regardless, the above method is still useful for symmetric matrices. --Adieyal 17:50, 29 July 2007 (UTC)

Is this the procedure called deflation? Anyway, feel free to add it in, preferably with a reference. It would be great if you could check the rest of the article and rewrite the parts where you can improve on it. -- Jitse Niesen (talk) 04:12, 30 July 2007 (UTC)

[edit] Broken links

I think the first reference to http://www4.ncsu.edu/~ipsen/ps/slides_imacs.pdf is broken. Please check. —Preceding unsigned comment added by 81.33.18.197 (talk) 09:31, 4 October 2007 (UTC)