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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!

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