# (Semi)Positive definiteness of product of symmetric positive (semi)definite matrices

Here is my problem. Any ideas are appreciated.

Let P be a projection matrix (symmetric, idempotent, positive semidefinite with 0 or 1 eigenvalues). For example, P = X*inv(X'*X)*X' where X is a regressor matrix in a least square problem.

Let A be a symmetric real matrix with only integer elements where the center submatrix (of a given size) is a (square, of course) matrix with identical elements, say 5. But the other elements of A are all smaller than the (common) element of the center submatrix (say, 5).

Q1: Is (P.*A)*P psd, nsd or indeterminant? where P.*A is the element-wise product of P and A (the Hadamard product)

Q2: Is P*(P.*A)*(I-P) psd, nsd or indeterminant? where I is the identity matrix of conformable size.

Comments: I have done some numerical examples in Matlab and it seems that the first matrix is psd and the second matrix has all zero eigenvalues (but not a zero matrix). Any idea as to how to prove the results?

I think I can show Q2 now. Q1 is still a puzzle. Any help is appreciated.
Also regarding the matrix A, does any one know of a theorem regarding the center submatrix of a matrix?

To put this simply, we know in general that if A and B are psd their product A*B is NOT necessarily psd.

Does anyone know when the product is indeed psd? I am looking for conditions on A and B to ensure the psd of their product.

Thanks a bunch

AB is not even necessarily symmetric. Consider the case where A and B commute (simple case A,B diagonal).