# Thread: Eigenvalues and Eigenvectors raised to a power.

1. ## Eigenvalues and Eigenvectors raised to a power.

If you have a matrix, and raise it to the power n, what would be the effect on the eigenvalues and the eigenvectors?

Thanks

If you have a matrix, and raise it to the power n, what would be the effect on the eigenvalues and the eigenvectors?

Thanks

$Av=\lambda v\Longrightarrow A^nv=\lambda^nv\,,\,\,\forall\,n\in\mathbb{N}$ . You can easily prove this by induction.

If the matrix is invertible the above is extendable to negative powers, too.

Tonio

3. Thank you,
Where can I find how to prove this?

Thank you,

Where can I find how to prove this?

In your head (honest: the proof is very easy), but I guess books in algebra or linear algebra must deal with this, though I think there's a good chance it'll be an exercise because, as already said, it is very easy. Have you REALLY tried to prove it by yourself already?

Tonio