From the point of view of linear algebra, the "natural" multiplication operation for matrices is the usual matrix product, and there are lots of theorems involving this product---e.g. the result $\det(AB) = \det(A)\det(B)$, or $\text{tr}(AB) = \text{tr}(BA)$, etc. However, there are lots of matrices one encounters in practice whose structure allows them to be written in a convenient way as an element-wise (Hadamard) product of two other matrices. This is one of the reasons why the default multiplication of arrays is element-wise in many programming languages (e.g. Python). In situations where element-wise products appear, it could be very nice to have theorems (like the above determinant & trace relations) concerning the linear algebraic character of the element-wise product. My question is: Do any "interesting" such theorems exist?
[I don't expect to find any results as slick as the above $\det$ and $\text{tr}$ identities, but perhaps there are analogous inequalities, or maybe some non-trivial statements about diagonalizability, or eigenvalue relations, etc.]
Yes, there is the Schur product theorem. See here: https://en.wikipedia.org/wiki/Schur_product_theorem
The Schur product or Hadamard product of two matrices is the elementwise product. The Schur product theorem states that the Schur product of two positive definite matrices is again positive definite.