Characteristic Polynomial

If A is a nxn matrix and is diagonalizable, consider the characteristic polynomial $\displaystyle p_A(t) $ and show that $\displaystyle p_A(A)$ is the zero matrix.

Can anyone help me out on the problem. I didn't that that the zero matrix could be diagonalizable.

Thanks!

Re: Characteristic Polynomial

??? The 0 matrix **is** already diagonal. A diagonal matrix is that has all 0s off the diagonal. It is quite allowable to have 0s on the diagonal also.

Actually, the theorem you have is more general than that: **every** matrix satisfies its own characteristic equation. However, for a diagonal matrix, it is easy to prove. The nth power of a diagonal matrix is just the diagonal matrix having the nth powers on it diagonal:

$\displaystyle \begin{bmatrix}2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 4\end{bmatrix}^n= \begin{bmatrix}2^n & 0 & 0 \\ 0 & 3^n & 0 \\ 0 & 0 & 4^n\end{bmatrix}$

It is also true that, for any matrix, A, and invertible matrix P, $\displaystyle (P^{-1}AP)^n= P^{-1}A^nP$.

Those are what you need.