I came across this matrix riddle a couple of weeks ago and I haven't figured it out.
- You have a known $10\times10$ matrix $A$, which is symmetric.
- For some unknown transformation matrix $T$, you have the relation that $\Lambda = T^TAT$.
$\Lambda$ is a diagonal matrix and $T$ is not necessarily orthogonal, so this isn't the normal diagonalization problem. - You have another relation for $T$: $J = TJT^T$, where $J$ is a $10\times10$ matrix such that the top right $5\times5$ block is an identity matrix and the bottom left $5\times5$ block is $-I$ (identity matrix).
How do you find what $T$ and $\Lambda$ are? Obviously $T$ is $10\times10$.
Using the two relations, I got the following equation: $J\Lambda = T^{-1}JAT$.
I don't if that helps.
Thanks!