# Matrix trace minimization and zeros

I would like to minimize and find the zeros of the function

$$F(S,P) = trace(S-SP^{T}(A+ PSP^{T})^{-1}PS)$$

in respect to $S$ and $P$.

$S$ is symmetric square matrix.

$P$ is a rectangular matrix

Could you help me? Thank you very much

All the best

GoodSpirit

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You have an implicit constraint that P and S should be such that $A+PSP^T$ is invertible. This will follow easily if A and S are assumed to be positive definite. IS it OK to make such additional assumptions? –  ACARCHAU Jan 23 '13 at 17:33
I really thank you for your answer.:) The fact is that A is positive definite and symmetric. I forgot to mention that... But S and P might might be positive semi-definite. I also would like to tell that the aim is to minimize an error metric and preferentially drive it to zero. Many thanks GoodSpirit –  GoodSpirit Jan 24 '13 at 12:13
Hello everybody, I've been using derivatives like this $dF/dS=0$ and $dF/dP=0$ What do you think? GoodSpirit –  GoodSpirit Jan 25 '13 at 12:06