EigenvaluesDate: 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/ |
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