I had been working on this problem here below, but seem to not know a precise and clean way to show the proof to the question below. I had about a few ways of doing it, but the statements/operations were pretty loosely used. The problem is as follows:

Suppose $\mathbf{A}$ is an $n \times n$ matrix with (not necessarily distinct) eigenvalues $\lambda\_{1}, \lambda\_{2}, \ldots, \lambda\_{n}$. Can it be shown that the *polynomial matrix* 

$p( \mathbf{A} ) = k\_{m} \mathbf{A}^{m}+k\_{m-1} \mathbf{A}^{m-1}+\ldots+k\_{1} \mathbf{A} +k\_{0} \mathbf{I} $ 

has the eigenvalues 

$p(\lambda\_{j}) = k\_{m}{\lambda\_{j}}^{m}+k\_{m-1}{\lambda\_{j}}^{m-1}+\ldots+k\_{1}\lambda\_{j}+k\_{0}$ 

where $j = 1,2,\ldots,n$ and the same eigenvectors as $\mathbf{A}$.

Thanks.