Eigenvalues and eigenvectors of Hadamard product of two positive definite matrices

Solution 1:

Generally speaking, no, there is no relationship among the eigenvalues/vectors of $A$, $B$, and their Hadamard product $A\circ B$. See, for example, the upvoted comment here.

I think the Wackypedia article you cite on the Schur product theorem has a nice section on how to use the eigenvalues/vectors of $A$ and $B$ to show $A\circ B$ is positive definite. It starts with knowing $A = \sum \alpha_i x_i x_i^T$ and $B = \sum \beta_i y_i y_i^T$. Then $$A \circ B = \sum_{ij} \alpha_i \beta_j (x_i x_i^T) \circ (y_j y_j^T) = \sum_{ij} \alpha_i \beta_j (x_i \circ y_j) (x_i \circ y_j)^T$$ Each term in the sum is positive semidefinite.

In your notation, $T$ is a matrix whose columns are the (normalized) eigenvectors of $A$, the $x_i$'s.