Talk:Power iteration

message: 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 \lambda_1 = 0. To see this we multiply by w_1
\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_2 w_1 \\
\end{align}
since w_1 and w_2 are orthogonal to each other.
--Adieyal 15:05, 5 July 2007 (UTC)
replies: text: Why are w_1 and w_2 orthogonal to each other? If I understand you correctly, w_1 and w_2 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)

text: 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)
thread_title:  Finding the 2nd Eigenvalue 

message: 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)
thread_title:  Broken links 

message: "However, it will find only one eigenvalue (the one with the greatest absolute value) and it may converge only slowly."
-------------------------------
I guess it is a matter of semantics, but this statement is not true for all
power methods. It is certainly true for the power method presented in the article,
but it is not true for the power method presented in
"Multiple extremal eigenpairs by the power method," Journal of Computational Physics, Volume 227 , Issue 19 (October 2008)
Maybe different names are needed for these slightly different "power methods?"
Certainly the statements:
"The power iteration is a very simple algorithm. It does not compute a matrix decomposition, and hence it can be used when A is a very large sparse matrix."
are true for the aforementioned alternate power method.
Nooseweak
-------------------------------
The lines:
    for(k=0; k<n; k++){
         norm_factor += tmp[k];
    }
are not calculating a vector norm - just look at the wikipedia definition of a norm.
I guess it should mean:
    for(k=0; k<n; k++){
         norm_factor += abs(tmp[k]);
    }
Which would refer to the 1-Norm (Manhattan-Norm)
Konan  —Preceding unsigned comment added by 93.134.111.116 (talk) 21:41, 17 May 2011 (UTC)
thread_title:  multiple eigenvalues by power method 

message: The algorithm illustrated by the code appears to be innacurate. The normalisation of the generated vector (which is the product of the matrix and the previous eigenvector) is done by finding the maximum element in the vector, and not by finding the magnitude of the vector. This does not affect the outcome of the power method in itself, but it can affect the outcome of variations of the power method, such as the Inverse Power Method, the Shifted Power Method, and the Shifted Inverse Power Method.
Sources:
> http://math.fullerton.edu/mathews/n2003/PowerMethodMod.html
> http://math.bard.edu/greg/math301/NumericalAnalysis.pdf
> http://macs.citadel.edu/chenm/344.dir/08.dir/lect4_2.pdf  — Preceding unsigned comment added by Krodh (talk • contribs) 12:39, 21 March 2012 (UTC)
thread_title:  The code presented is inaccurate 

message: The proof given in the analysis section does not use this assumption.  — Preceding unsigned comment added by 73.42.179.192 (talk) 02:51, 26 November 2020 (UTC)
thread_title:  Why does the introduction have the assumption that the matrix is diagonalizable?