Complex and real counterexamples to an inequality arising from Drury's conjecture

Abstract

We give counterexamples to the following inequality, stated by Zhang as a consequence of Drury's 2016 permanent conjecture: if A = (ajk)j,k=1n is a Hermitian positive semidefinite matrix of order n ≥ 2, then ∑j=1n−1 |ajn|2 per A[ĵ, n̂] ≤ per A, where A[ĵ, n̂] is the principal submatrix obtained by deleting indices j and n. Our complex example has order twelve and rank two; our real example has order twenty-one and rank three.