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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?