1
$\begingroup$

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!

$\endgroup$
  • $\begingroup$ Why doesnt the LaTeX code show up? This is very irritating, and especially the code in this question is about impossible to read! $\endgroup$ – kjetil b halvorsen Sep 10 '12 at 19:01
  • $\begingroup$ 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. $\endgroup$ – rschwieb Sep 10 '12 at 19:14
  • $\begingroup$ 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? $\endgroup$ – rschwieb Sep 10 '12 at 19:17
  • $\begingroup$ I gather there are mathjax issues at the moment due to some godaddy problems... $\endgroup$ – copper.hat Sep 10 '12 at 19:22
  • $\begingroup$ I can see it now, btw. I must have done something right... Or else I just messed up the whole question :) $\endgroup$ – rschwieb Sep 10 '12 at 20:14
3
$\begingroup$

No.

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 $T^{-1}\neq T^T$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.