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