2
$\begingroup$

I have a matrix $A$ that is known to be positive definite, and I want to conclude something about the matrix whose entries are the square of $A$'s entries (each entry is the square of the corresponding entry, I am not talking about the matrix product $A\times A$). Can something be concluded?

About $A$: it is symmetrical, its diagonal entries are all 1 and the off-diagonal entries are all $\ge0$ and $<1$. The matrix is not diagonally dominant (I am actually dealing with a family of matrices, some are diagonally dominant, so that the conclusion is easy, but some are not).

More info: $A$ is actually the Gram matrix of $n$ vectors all of norm 1, so that $a_{ij}=x_i\cdot x_j=\cos\theta_{ij}$, where $\theta_{ij}$ is the angle between the two vectors. All vector components are $\ge0$ and $\le1$.

$\endgroup$
5
$\begingroup$

Yes, the entrywise square of $A$ remains positive definite if $A$ is positive definite. This is knwon as the Schur product theorem. The Wikipedia article contains a proof that makes use of covariance matrices, which may be relevant to your study.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.