Eigenvalues and Eigenvectors raised to a power.

• Mar 20th 2010, 07:56 AM
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
• Mar 20th 2010, 11:24 AM
tonio
Quote:

Originally Posted by llamagoogle
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
• Mar 20th 2010, 12:22 PM
Thank you,
Where can I find how to prove this?
• Mar 20th 2010, 01:55 PM
tonio
Quote:

Originally Posted by llamagoogle
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