quote:Original post by Vlion
Ax = cA, ya, ya, thats right.

No it''s not. The eigenvalues,c, for the matrix A satisfy the equation: Ax = cx , as etaylor27 pointed out correctly.

For each eigenvalue, ci, there exists a corresponding eigenvector. As has been pointed out, the eigenvectors of a matrix A represent an orthogonal basis for the space spanned by A.

How else are eigenvalues and eigenvectors useful?

In dynamic systems, the eigenvalues of the phase space matrix relate the divergence (+ve values) and convergence (-ve values) of neighbouring trajectories.

For any matrix A, the eigenvalues represent the amplitute of each mode of the matrix (as the AP mentioned)... and are thus related to such descriptions as Fourier Series.

Eigenvalues and eigenvectors can be used to find solutions for differential equations.

They find usage in approximation and filtering techniques. By re-expressing a matrix using only a selected subset of its eigenvalues, we can filter out modes of our choice from the system.

There are many, many more uses for eigenvalues and eigenvectors. Too many for me to describe in one post (and certainly more than I actually know about)! Get a decent book on linear algebra and do a lot of reading in the library!

Cheers,

Timkin