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.]
Best Answer
A matrix $A$ is called doubly nonnegative (DN) if it is entrywise nonnegative and positive semidefinite. For $A \in M_n (\mathbb{C})$ and $\alpha \in \mathbb{R}$, denote by $A^{(\alpha)}$ the entryise Hadamard power, i.e., $A^{(\alpha)} = [a_{ij}^\alpha]$.
Let $A$ be a DN matrix. Horn and Fitzgerald [MR0506356; J. Math. Anal. Appl. 61 (1977), no. 3, 633–642] showed that $A^{(\alpha)}$ is DN if and only if $\alpha \in \mathbb{N} \cup [n-2,\infty)$. The methods in this paper are also used to give another proof of the Schur product theorem (cited in another answer).
Interestingly, this lower bound is the same for conventional matrix powers; Johnson et al. [MR2810562; Linear Algebra Appl. 435 (2011), no. 9, 2175–2182] established the existence of a critical exponent and conjectured that $A^\alpha$ is DN for every $\alpha \ge 2$. Guillot et al. [MR3091314; Linear Algebra Appl. 439 (2013), no. 8, 2422–2427] settled the conjecture in the affirmative.
For $A \in M_n(\mathbb{C})$, denote by $\rho(A)$ the spectral radius of $A$. It is known that the spectral radius is sub-multiplicative with respect to the Hadamard product ($\odot$) for non-negative matrices; i.e., $$ \rho(A \odot B) \le \rho(A)\rho(B),~\forall A,B \ge 0. $$ The standard reference for this is Topics in matrix analysis by Horn and Johnson.