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