Semi-Positive Definiteness of Product of Symmetric Matrices

[iamhappy]

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?

 

[iamhappy]

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

 

[iamhappy]

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

 

[trambolin]

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

 

[blue_raver22]

I can provide some insights and possible approaches to your problem.

Firstly, let's define the terms used in your question. A symmetric matrix is a matrix that is equal to its transpose, i.e. A = A^T. A projection matrix is a matrix that when multiplied with itself, gives the same matrix, i.e. P^2 = P. Positive semidefinite matrices have all non-negative eigenvalues, while negative semidefinite matrices have all non-positive eigenvalues. Indefinite matrices have both positive and negative eigenvalues.

Now, let's address your first question. The product of two symmetric matrices is also a symmetric matrix. Therefore, (P.*A) is symmetric. Also, since P is positive semidefinite and A has only integer elements, the element-wise product (P.*A) will also have non-negative elements. This means that (P.*A) is positive semidefinite.

Since the product of two positive semidefinite matrices is also positive semidefinite, (P.*A)*P is also positive semidefinite. Therefore, the answer to your first question is that (P.*A)*P is positive semidefinite.

For your second question, let's first consider the matrix P*(P.*A). Since both P and (P.*A) are positive semidefinite, their product will also be positive semidefinite.

Now, let's look at the matrix (I-P). Since P is a projection matrix, (I-P) is also a projection matrix. This means that (I-P)^2 = (I-P). Therefore, (I-P) is also a projection matrix.

Now, let's consider the product (P.*A)*(I-P). Since (P.*A) is positive semidefinite and (I-P) is a projection matrix, their product will also be positive semidefinite.

Finally, we have P*(P.*A)*(I-P). Since P and (P.*A)*(I-P) are both positive semidefinite, their product will also be positive semidefinite.

Therefore, the answer to your second question is that P*(P.*A)*(I-P) is positive semidefinite.

To prove these results, you can use the properties of positive semidefinite matrices and projection matrices, as well as the fact that the product of two positive semidefinite matrices is also positive semide