Finding a transformation matrix $\hat{T}$ such that $\hat{T}A \hat{T}^{-1}$ is diagonal?

Question

Given an example matrix $A$ as follows: $$A = \begin{bmatrix} 3 & 1 & -1 \\ -1 & 1 & 1 \\ 2 & 2 & 0 \end{bmatrix} $$

I know how to find the matrix $T$ such that $T^{-1}AT$ is diagonal. ie: You simply find the eigenvectors and you are good to go.

Question:

What if I wanted to find the matrix $\hat{T}$ such that $\hat{T}A \space\hat{T}^{-1}$ is diagonal? Is this related to $T$ in any way? That is, can I still use the eigenvectors of $A$ in some way to create this matrix? I feel like I'm missing something obvious as this doesn't seem like a difficult task.

Thanks!

Answer 1

If $T^{-1}AT$ is diagonal and you want $\hat{T}A\hat{T}^{-1}$ to be diagonal, then just take $\hat{T}=T^{-1}$.