? eigenvalue prob

Hi:

In finite dimension, an eigenvalue problem is given as

A*x = lambda*x, where A is a N-by-N matrix and x is a

N-vector. The existence of eigenvectors is assured by a

fundamental theorem. (Well, something realted to complex

roots of a polynominal but I forget its name though.)

It is also known that eigenvectors of a given N-by-N matrix

need not mutually orthogonal to each other.

There are also infinite dimensional eigenvalue problem.

But, I only learnt someting very limited in this topic; only

that one can solve PDEs using Fourier series or Fourier transform.

When one is solving PDEs using those methods, one deals with

int-dim eigenvalue problem.

My questions now are: 1) how does one assure the existence

of eigenfunctions defined by PDEs, since a set of PDEs with BCs

need not always have a solution, and 2) are there also non-orthogonal

eigenfunctions as one faced in finite-dim eigenvalue problem? If so,

is orthogonal eigenfunction more general or non-orthogonal eigenfunction

more general? Can someone give an example for an eigenvalue problem

that has non-orthogoanl eigenfunctions?

Thanks,
by Cheng Cosine
Mar/23/2k8 NC


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