Diagonalizable Matrices: What Values of a Make This Matrix Diagonalizable?

[lineintegral1]

Hey all, I have a question on this specific application of diagonalizable matrices.

Homework Statement

For what values of the real constant a is the matrix diagonalizable over $$\mathbb{C}$$? For what values is the matrix diagonalizable over $$\mathbb{R}$$?

$$\begin{bmatrix}<br /> 0 & 0 & a\\ <br /> 1 & 0 & 3\\<br /> 0 & 1 & 0<br /> \end{bmatrix}$$

Homework Equations

N/A

The Attempt at a Solution

I assume that you compute the eigenvalues and check to see for what values of a do the geometric multiplicities equal the algebraic multiplicities. But the characteristic polynomial is a bit messy since it has that arbitrary constant. Am I missing something dumb? Can anyone offer some insights as to how to put these ideas together? Thanks!

- Zach

 

[Dick]

Sure, you don't want to actually solve the eigenvalue equation, but if the characteristic polynomial f(x) has a multiple root at say x=b, then f'(b)=0, right? Exploit that.