Sufficient conditions for a sum of Hadamard products of positive semidefinite matrices to be positive definite

Question

Let $A_i$, $B_i$ be hermitian $n$ by $n$ positive semidefinite matrices of rank $1$, for $i = 1, \dots, n$. Assume that the rank of $A_i \circ B_i$ is also $1$, for $i = 1, \dots, n$, where $\circ$ denotes the Hadamard product. We have that $$A_1 + \dots + A_n > 0$$ (i.e. the LHS is positive definite). I am interested in finding sufficient conditions on the $A_i$ and $B_i$, for $$ C := A_1 \circ B_1 + \dots + A_n \circ B_n $$ to be positive definite. What kind of sufficient conditions? I guess the weaker the better, so that hopefully my specific $A_i$ and $B_i$ would satisfy them...

Answer 1

Writing $A_i = u_iu_i^\dagger$ and $B_i = v_iv_i^\dagger$, your assumption that $A_i \circ B_i$ has rank $1$ is equivalent to $u_i \circ v_i \ne 0$, and your assumption that $\sum_i A_i \succ 0$ is equivalent to $\operatorname{Span}_\mathbb{C}\{u_i\} = \mathbb{C}^n$, or equivalently $\det(u_1, ..., u_n) \ne 0$.

Your desired conclusion $\sum_i A_i \circ B_i \succ 0$ is equivalent to $\operatorname{Span}_\mathbb{C}\{u_i \circ v_i\} = \mathbb{C}^n$, or to $\det(u_1 \circ v_1, ..., u_n \circ v_n) \ne 0$. (Note that $u_i \circ v_i$ is proportional to every nonzero column of $A_i \circ B_i$, so this is also equivalent to the collection of all columns occurring among all of the matrices $A_1 \circ B_1, ... A_n \circ B_n$ having a full-dimensional span.)

A few observations: first, the set of tuples $v_1, ..., v_n$ such that $\det(u_1 \circ v_1, ..., u_n \circ v_n) \ne 0$ is open in the Zariski topology, so if the $v_i$ are chosen at random you will succeed with probability $1$. Second, it's easy to construct examples where it fails, for instance, if $u_1 \circ v_1$ and $u_2 \circ v_2$ are proportional to each other (which will occur if, say, $v_1 = u_2$ and $v_2 = u_1$) then you will be cooked.

Anyway, this has all boiled down to looking for sufficient conditions for a square matrix to have a nonzero determinant. For instance, you could try checking whether the matrix $(u_1 \circ v_1, ..., u_n \circ v_n)$ is strictly diagonally dominant (or whether its transpose is strictly diagonally dominant), perhaps after a clever change of basis.