Matrix riddle I came across

I came across this matrix riddle a couple of weeks ago and I haven't figured it out.

1. You have a known $10\times10$ matrix $A$, which is symmetric.
2. 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.
3. 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!

• Are the top left and bottom right blocks of $J$ unknown? – joriki Sep 17 '12 at 7:30
• They are all zeros. – user34801 Sep 17 '12 at 18:26
• @Gerry Myerson: I'm sorry, but I actually didn't know that you could accept answers until you and someone else pointed it out to me. – user34801 Sep 17 '12 at 18:31
• Have you checked the $2\times2$ case, $J=\pmatrix{1&0\cr0&-1\cr}$? Maybe small enough to write everything out explicitly to see what's going on, and then see what applies to the $10\times10$. – Gerry Myerson Sep 18 '12 at 1:22
• Yeah, so I am not able to solve it for a 2x2 case either because I don't know what T or lambda are. The only thing I know is that Jlambda and JA are related by a similarity transform, the transformation matrix being T. But equating their eigenvalues gives me just one 5 equations... So, I'm not sure how to proceed. – user34801 Sep 18 '12 at 3:21