Semi-Positive Definiteness of Product of Symmetric Matrices

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Discussion Overview

The discussion revolves around the semi-positive definiteness of products involving symmetric matrices, specifically focusing on the Hadamard product of a projection matrix and another symmetric matrix. The participants explore conditions under which certain matrix products are positive semidefinite (psd), negative semidefinite (nsd), or indeterminate.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant presents a problem involving a projection matrix P and a symmetric matrix A, asking whether the Hadamard product (P.*A)*P is psd, nsd, or indeterminate.
  • The same participant queries the semi-positive definiteness of the product P*(P.*A)*(I-P) and notes numerical examples suggesting the first matrix is psd and the second has all zero eigenvalues.
  • Another participant asserts that the product of two psd matrices A and B is not necessarily psd and seeks conditions under which their product is psd.
  • A further comment highlights that the product AB is not necessarily symmetric, using the case of commuting diagonal matrices as an example.

Areas of Agreement / Disagreement

Participants express uncertainty regarding the semi-positive definiteness of the matrices in question, with some agreeing on the complexity of the conditions required for psd products, while no consensus is reached on the specific cases presented.

Contextual Notes

Participants mention the need for further exploration of theorems related to the center submatrix of a matrix and the conditions under which products of symmetric matrices retain positive semidefiniteness. The discussion reflects a reliance on numerical examples and theoretical inquiry without definitive conclusions.

Who May Find This Useful

Researchers and students interested in matrix theory, particularly in the properties of symmetric matrices and their products, may find this discussion relevant.

iamhappy
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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?
 
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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?
 


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).
 

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