An analytic approach to a permanent conjecture

This paper is concerned with the conjecture on the permanent of the Hadamard product of two positive semidefinite matrices: . We present some properties of correlation matrices and introduce an analytic approach of maximizing matrices for the permanent conjecture. Given an irreducible correlation matrix, we show that its maximizing matrix is (i) singular, (ii) irreducible, and (iii) invariant by a row and column reduction.