Semi-Positive Definiteness of Product of Symmetric Matrices

In summary, the conversation discusses the properties of projection matrices and symmetric real matrices with integer elements. The question is raised about whether the element-wise product of a projection matrix and a symmetric real matrix is positive semidefinite, negative semidefinite, or indeterminant. The second question asks about the properties of the product of a projection matrix, the element-wise product of the projection matrix and a symmetric matrix, and the identity matrix. The conversation also mentions the search for a theorem regarding the center submatrix of a matrix and conditions for the product of two positive semidefinite matrices to also be positive semidefinite.
  • #1
iamhappy
3
0
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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  • #2


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?
 
  • #3


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
 
  • #4


AB is not even necessarily symmetric. Consider the case where A and B commute (simple case A,B diagonal).
 
  • #5


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
 

1. What is meant by "Semi-Positive Definiteness"?

Semi-Positive Definiteness refers to a property of a matrix where all of its eigenvalues are either non-negative or equal to zero. It indicates that the matrix is positive semidefinite, which means that it has no negative eigenvalues and all of its principal minors are non-negative.

2. How is the Semi-Positive Definiteness of a product of symmetric matrices determined?

The Semi-Positive Definiteness of a product of symmetric matrices can be determined by checking the eigenvalues of the product matrix. If all of the eigenvalues are non-negative or equal to zero, then the matrix is semi-positive definite.

3. Can a product of two symmetric matrices be semi-positive definite if one of the matrices is not symmetric?

No, in order for a product of two matrices to be symmetric, both of the matrices must also be symmetric. Otherwise, the product matrix may not have the same properties as a symmetric matrix and may not be semi-positive definite.

4. What is the significance of the Semi-Positive Definiteness of a product of symmetric matrices?

The Semi-Positive Definiteness of a product of symmetric matrices is important in many areas of mathematics and science, including optimization, control theory, and statistics. It allows for the use of efficient algorithms and mathematical techniques to solve problems involving these matrices.

5. Are there any other types of definiteness for matrix products besides Semi-Positive Definiteness?

Yes, there are two other types of definiteness for matrix products - Positive Definiteness and Negative Definiteness. Positive Definiteness refers to a matrix with all positive eigenvalues, while Negative Definiteness refers to a matrix with all negative eigenvalues. These types of definiteness can also apply to products of symmetric matrices.

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