Eigenvalues

Date: 12/18/98 at 08:25:17
From: Mike Zuber
Subject: Eigenvalues

What is an eigenvalue and how is it used?

I've looked this up in various math texts. Unfortunately, the 
explanations were a bit recondite. I'm looking for general information 
so I have at least a rudimentary understanding when this topic comes up 
in work.

Thank you.

Date: 12/18/98 at 09:37:40
From: Doctor Mitteldorf
Subject: Re: Eigenvalues

Dear Mike,

This is one of those topics about which it is difficult to imagine how 
it could be useful until the day when you need it, when it's just the 
thing.

Start with a square n*n matrix and try multiplying it by different 
vectors. You'll find, in general, that after multiplication the vectors 
point in a different direction than before. The matrix "rotates" the 
vector to a different direction in n-space. 
 
But there are certain vectors that remain fixed in direction. They are 
stretched or shrunk, but they still point in the same direction after 
they're multiplied by the matrix. These are "eigenvectors."  The 
"eigenvalue" is the ratio of the length of the new vector to the length 
of the old, i.e., the degree to which the un-rotated vector has been 
stretched by the action of the matrix.

I come from the world of physics, where there are two common uses for 
eigenvectors and eigenvalues. First, in quantum mechanics, eigen-
problems are ubiquitous. Schroedinger discovered 70 years ago that the 
quantum state of an atom or particle or any system is an eigenfunction 
of the energy operator. Therefore much of what the quantum physicist 
does is to write down energy operators (called Hamiltonians) 
appropriate to a given situation, and look for eigenvectors. The 
eigenvalues are the possible energies of the system, and the 
differences among the eigenvalues give the spectrum of emission of 
absorption.

The second physics application that comes to mind is stress and strain 
deformation of crystals. A stress on a crystal will deform it, and in 
general the deformation can be in a different direction from the 
stress. But there are 3 directions of stress in which the deformation 
is parallel to the stress pressure. Sounds like an eigenvector problem.

I'd recommend you approach the problem from the point of view of an 
application. The mathematics of eigenvectors can seem awfully abstract 
if you don't have an application in mind.

- Doctor Mitteldorf, The Math Forum
  http://mathforum.org/dr.math/