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 $\le1$.
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.