Need to find eigenvector that corresponds to max eigenvalue

[jpildave]

To give you some background, I am trying to perform an AHP calculation using Java code. I have a 15x15 matrix and I need to find its eigenvector. I want the eigenvector that corresponds to the greatest eigenvalue.

Let's say I already have some method that gives me all the eigenvectors and all the eigenvalues. Let's call the eigenvector matrix V (15x15) and the eigenvalues matrix D (15x15 block diagonal).

I'm confused because some of my eigenvalues are complex and so I don't exactly know how to find the maximum one. I believe I have to do something like a^2 + b^2, but this is where I need help. Do I do this per column in D? (I need the exact steps in calculating and then also find the maximum one.)

Thank you.

 

[HallsofIvy]

Since the complex numbers are not an ordered field, I assume you mean the largest real eigenvalue. Algorithms for finding eigenvalues are much to complicated to give in detail here but here is a website about using the QR algorithm to find the largest eigenvalue of a matrix:
http://en.wikipedia.org/wiki/QR_algorithm
and here is information about the power and Lanczos algorithms:
http://www.nationmaster.com/encyclopedia/Lanczos-algorithm

 

[jpildave]

HallsofIvy,

Thanks for your response. I wanted to follow-up on something you mentioned... You said that you think I should be looking for the largest real eigenvalue. I didn't know that I should only be limiting the search to the largest _real_ eigenvalue. I think this will simplify things for me, but can you explain why I only need to be concerned with the _real_ eigenvalues? (Sorry, if this is a stupid question.)

Since I already have the eigenvalues and and the eigenvectors, I do not need to code the QR algorithm myself, however, I have an additional question.

In the 15x15 matrix that contains the eigenvalues, does the following logic make sense to you to find the largest real eigenvalue...?

Look at each column in the matrix. If there are 2 non-zero values in the column, then this is an eigenvalue that is complex. Therefore, don't worry about it and go to the next one. But, if you come find a column that only has one non-zero value, this represents an eigenvalue that is real. Therefore, take note of this value and proceed.

Sound good?