# Math Help - Eigenvalue and matrices

1. ## Eigenvalue and matrices

Hi,
I want to show that a matrix H that has an eigenvalue t can be used to show that the matrix $\textbf{H}^n$ would have an eigenvalue of $t^n$.

The formal way of doing this i would guess to be mathematical induction, though how would i do the inductive step? (As i couldn't take the determinant of the whole thing)

thanks

2. ## Re: Eigenvalue and matrices

If $x$ is an eigenvector that has eigenvalue $t$, then $Hx=tx$. Therefore, $H^{n+1}x=H(H^nx)=H(t^nx)=t^n(Hx)=t^{n+1}x$.