# Eigen Funtions

what is a eigen value and eigen function? i have read a lot abt it...i understand the math behind it.. what is its physical significance of it?

HallsofIvy
Have you studied Linear Algebra? That's really where it comes from. If A is a linear operator on some vector space, the "eigenvalue problem" is: $Av= \lambda v$. That has, of course, the "trivial" solution v= 0. For some values of $\lambda$, called "eigenvalues", there are other, non-trivial, solutions- in fact the set of all solutions, in that case, is a sub-space. Those non-trivial solutions are the "eigenvectors".
Of course, in the finite dimensional case, after we have chosen a basis, we can write a linear operator as a matrix, with each column showing what the operator takes each basis vector to (written in terms of that basis). If we choose an eigenvector, corresponding to eigen value $\lambda$ as the "nth" basis vector, the nth column consists of "0"s except for the value $\lambda$ in the nth row. In particular, if we can find a "complete set of eigenvectors"- that is, a basis consisting entirely of eigenvectors- which we always can in the important "self adjoint operator" case, then the matrix representing the linear operator in that basis is diagonal- the simplest kind of matrix.