Eigenvalues and Eigenvectors

Does anyone know a site that provides an algorithm for finding this(eigenvalues).

What I am trying to do: Compute the eigenvalue(s) of an n x n matrix.


Questions: If lambda = x then det(A-xI)=0, How do I make a matrix multiply and and subtract nonreal numbers in excel. So if I have a 4x4 matrix by using det(A-xI)=0 I will get an equation that looks like ax^4+bx^3+cx^2+dx+e=0 where a, b, c, d, e are some "real numbers". But how do I make excel spit that out? Is this possible in excel?

Alternative ways of producing eigenvalues?

Math is not my issue, I just don't know what excel can and cannot do, and how to do it. YET!

It's been 10+ years since I was in college, so I had to dig out my diff equations text to remind myself what eigenvalues and eigenvectors are.

As you no doubt know, eigenvalues are the roots of a polynomial equation as you described. The most obvious utility for finding roots of polynomials in Excel is the Solver utility. Solver uses the Newton-Raphson method solve various types of problems. Basically you will set up a spreadsheet to calculate y=f(x) at some x. Then ask Solver to find an x that corresponds to y=0.

A couple of issues. I've never used complex numbers in Excel. If you only need real solutions, I doubt you'll have any trouble. However, if you do need complex solutions, Excel does have several functions for doing basic operations with complex numbers (most of these functions begin with IM...(). See Excel Help's list of Engineering functions). There's a good chance you'll be able to get Excel/Solver to find complex solutions as well.

2nd issue: Do you want to find an eigenvalue or all of the eigenvalues? One of the problems when using numerical methods to find roots of polynomials (and higher order polynomials require numerical methods to get the roots) is knowing when you have found all of the roots, especially when multiple order roots exist. In the case of a 4th order polynomial like you've proposed, there are 4 roots. If I start with 4 different initial values and I converge on 4 unique solutions, obviously I know I've found all four solutions. However, if I start with 4 different initial values, and the algorithm converges on 2 unique solutions, how do I know if either or both of those is a multiple order root, or, put another way, how do I know I have found all of the solutions?

Excel doesn't have a built in utility that will know this for you. You, as the programmer, are going to have to devise an algorithm that will tell you if you've found all the roots. I don't know how familiar you are with numerical methods, or implementing numerical methods on computers. If you're not familiar with these kinds of algorithms, I would suggest you find a good numerical methods text that will help with these kinds of issues.

Once you have the eigenvalues, obtaining the eigenvectors should be straightforward.