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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.

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    $\begingroup$ Possibly helpful: math.stackexchange.com/q/23312/215011 $\endgroup$ Commented Apr 7, 2015 at 18:46
  • $\begingroup$ @grand_chat - Thank you so much! :). That really is an AMAZING explanation! $\endgroup$ Commented Apr 7, 2015 at 18:50

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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.

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