(For this queation, all matrices are real).
According to the ancient paper http://www.springerlink.com/content/l455p582210k1113/ which I cannot really read fully, since it is in german, any square matrix can be written as a product of two symmetric matrices (one of which is non-singular). If we strengthen the conditions, such that one of the factors must be positive-(semi)definite, what can we say? Any way of characterizing square matrices which can be written as a product of a symmetric and a symmetric positive-semidefinie positive-semidefinit matrix?
If $A$ is symmetric positive-definite and $B$ is symmetric, then the product $AB$ is similar to a symmetric matrix, so has real eigenvalues. So if any square matrix could be written such, all square matrices would have real eigenvalues, which is absurd. So there must be some restriction.
And, any more recent references for this problem?