For which complex parameters the following matrix is diagonalizable

Question

For all possible complex values of the parameter $\lambda$, determine if the matrix $A$ is diagonalizable and if so find an invertible matrix $C$ and a diagonal matrix $D$ so that $C^{-1}$$DC=A$

$A$ = $\left( \begin{array}{ccc} 1 & i & 0 \\ 0 & \lambda & 1 \\ 0 & 0 & i \end{array} \right)$

To begin with, I am not even sure what are the parameters I should be working with for $\lambda$ and then how should I proceed with them?

Answer 1

Note that the characteristic polynomial of your matrix is $$ (i-x)(1-x)(\lambda-x)=0 $$ and, if $\lambda \ne i$ and $\lambda \ne 1$ it has three distinct eigenvalue: $x=1$ , $x=i$ and $x=\lambda$.

This is a sufficient condition for the matrix to be diagonalizable and the similar diagonal matrix is $$ \begin{bmatrix} i&0&0\\ 0&1&0\\ 0&0&\lambda \end{bmatrix} $$

Now find the eigenvectors for these eigenvalue and they are the columns of the matrix $C^{-1}$

Answer 2

A matrix $A\in F^{n\times n}$ is diagonalizable iff the sum of the dimensions of eigenspaces is equal to $n$. $C$ is then given in terms of eigenvectors.

Hence, all you need to do is to compute the dimensions of the eigenspaces of your matrix depending on the parameter $\lambda$ and (if the criterion applies) compute a basis of eigenvectors.

I would suggest you distinguish the following cases $\lambda\notin\{ 1, i\}$ (easier case) and $\lambda \in \{1, i\}$ (more difficult).