# Eigenvalue Problem

• I
While reading problems in my physics book , I encountered a statement very often "Eigen Value Problem" , I read about it from many sources , but wasn't able to understand it . So what exactly is an Eigen Value Problem?

Strilanc

That's a long page and the only mention of the Eigen Value problem seems to be right at the beginning where it says "re-directed from Eigen Value problem".

Let A=Any vector, x=eigen vector, e=eigen value
The definition of eigenvalue is the following
Ax=ex [where x=eigen vector corresponding to this value],this allows us to find particular values e whereby we can map the vector A into a multiple of itself.

Let A=Any vector, x=eigen vector, e=eigen value
The definition of eigenvalue is the following
Ax=ex [where x=eigen vector corresponding to this value],this allows us to find particular values e whereby we can map the vector A into a multiple of itself.

## A ## is not a vector. It is a linear operator.

The eigenvalue problem is: given a linear operator ## A ## (in a matrix form or otherwise), find it eigenvalues ## \lambda ## and eigenvectors ## u ## defined as $$Au = \lambda u.$$ Intuitively, an eigenvector is a vector that does not change its direction (except when ## \lambda ## is negative it got flipped) upon the action of ## A ##. Examples of eigenvalue problems ubiquitous in physics are finding normal modes of wave equations or stationary states of the Schrödinger equation in quantum mechanics. In these examples, ## \lambda ## are frequencies and energies respectively. They are eigenvalue problems because differential operators, ##\frac{d}{dx}## and linear combinations of its powers, are linear operators: the derivative of the sum is the same as the sum of derivatives.

kith