Eigenvalue Problem: What Is It?

[Hamza Abbasi]

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]

https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors

 

[cosmik debris]

https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors

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".

 

[whit3r0se-]

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.

 

[Truecrimson]

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]

In order to get a feel for what eigenvalues and eigenvectors are, it is very instructive to look at linear transformations in two dimensions. Are you familiar with the basics of linear algebra (vectors, matrices, changing bases, etc.)?

 

[cosmik debris]

I think the OP was asking what the problem is not what Eigenvectors and values are. I think the problem is that it is problem not problem. Damn english!