linear algebra - What are eigenvalues and eigenvectors really?

I know how to determine the eigenvalues and eigenvectors of a given matrix $A$, but we were not really explained to what exactly ARE eigenvalues and eigenvectors, what is their purpose and what exactly do/can they tell us about a matrix/system?

Can someone please provide me with some information about this? It will be much appreciated.

Answer

If $A$ is an $n \times n$ matrix, the nonzero $n$-component column vector $x$ is an eigenvector for eigenvalue $\lambda$ if $A x = \lambda x$.

See e.g. Wikipedia which discusses many of the uses of these.