Matrix trace minimization and zeros

Hello, :)

I would like to minimize and find the zeros of the function F(S,P)=trace(S-SP’(A+ PSP’)^-1PS) 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
 PhysOrg.com science news on PhysOrg.com >> Galaxies fed by funnels of fuel>> The better to see you with: Scientists build record-setting metamaterial flat lens>> Google eyes emerging markets networks
 Hello everybody, Perhaps I should explain a little bit. The aim is to minimize an error metric and preferentially drive it to zero. This should be done as function of S and P, as function of their rank and dimensions in particular. By the way, the matrix A is symmetric too. Many thanks
 Hello, Trying to update the equation presentation. $$F(S,P)=tr(S-S P^T(A+PSP^T)^-1 PS)$$ A is positive definite I've using matrix derivatives What do you think? All the best GoodSpirit

Matrix trace minimization and zeros

LateX didn't work here

How to present an equation here?

Thank you

Good Spirit