An extremal property of the permanent and the determinant

Abstract

Given an n×n matrix A, define the n!×n! matrix Ã, with rows and columns indexed by the permutation group S_n, with (σ,τ) entry Πⁿ_i=1 a _{τ(i), σ(i)}. It is shown that if A is positive semidefinite, then det A is the smallest eigenvalue of Ã; it is conjectured that per A is the largest eigenvalue of Ã, and the conjecture proved for n^≤3. Several known and some unknown inequalities are derived as consequences.