Matrix Diagonalization

I have a matrix equation that reads: $T^{-1} A ( T^{\mathrm{T}} )^{-1} =D$ where $D$ is a diagonal matrix.

Can I conclude that $T^{-1} = T^{\mathrm{T}}$ and that $T$ is the matrix of the eigenvectors of $A$?

Thanks!

• Why doesnt the LaTeX code show up? This is very irritating, and especially the code in this question is about impossible to read! – kjetil b halvorsen Sep 10 '12 at 19:01
• I tried a few simplifications to try to get it to go, but it doesn't seem to want to budge... For example, I tried \mathsf in a field elsewhere, and it didn't render, so that's gone. – rschwieb Sep 10 '12 at 19:14
• By "can I do <random thing>" I guess you are wishing that this would work. Perhaps the problem that led you up to this would be useful? Along with any work you've already done? – rschwieb Sep 10 '12 at 19:17
• I gather there are mathjax issues at the moment due to some godaddy problems... – copper.hat Sep 10 '12 at 19:22
• I can see it now, btw. I must have done something right... Or else I just messed up the whole question :) – rschwieb Sep 10 '12 at 20:14

Let $D$ be the identity matrix, $T=\begin{bmatrix}1&1\\0&1\end{bmatrix}$ and $A=\begin{bmatrix}2&1\\1&1\end{bmatrix}$.
Then you can compute that $TDT^T=A$, but $T^{-1}\neq T^T$.