What are eigenvalues and eigenvectors really?

Question

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.

Possibly helpful: http://math.stackexchange.com/q/23312/215011 — grand_chat, Apr 07 '15 at 18:46
@grand_chat - Thank you so much! :). That really is an AMAZING explanation! — user860374, Apr 07 '15 at 18:50

score 3 · Accepted Answer · answered Apr 07 '15 at 18:34

3

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.

answered Apr 07 '15 at 18:34

Robert Israel

448,999

What are eigenvalues and eigenvectors really?

1 Answers1