Let $B$ be a fixed symmetric $M\times M$ matrix over the reals.

Let $A$ be an arbitrary $N\times M$ matrix over the reals.

I want to consider the problem of finding the extremal value of $\operatorname{Tr}(ABA^T) = \operatorname{Tr}(A^TAB)$ under the constraint that $AA^T$ is a fixed $N\times N$ matrix.

Can you give me a hint?

`$\sum \lambda_i \mu_i$`

where`$\lambda_1 \geq \lambda_2 \geq \dots$`

(resp.`$\mu_1 \geq \mu_2 \geq \dots$`

) is the non-decreasing sequence of eigenvalues of $BB'$ (resp. $AA'$). – Mikael de la Salle Aug 2 '11 at 10:34