Mixed discriminants of positive semidefinite matrices

Abstract

If A^k=(a^k_ij), k= 1,2,…,n, are n-by-n matrices, then their mixed discriminant D(A¹,…,Aⁿ) is given by D(A¹.....Aⁿ=1/n! Σ_σ∈S_a| (α_ij^α_(j))| where S_n is the symmetric group of degree n and where |·| denotes determinant. We give certain alternative ways of defining the mixed discriminant and state some basic properties. It is pointed out that a Ryser-type formula for the mixed discriminant exists in the literature, and a simpler proof is given for it. It is shown that the mixed discriminant can be expressed as an inner product. A generalization of Konig's theorem on 0-1 matrices is proved. The following set D_n, which includes the set of n-by-n doubly stochastic matrices, is defined and studied: D_n={(A¹.....,Aⁿ):Aⁱ is a-by-n, positive semidefinite with trace 1, i=1,2,....n; Σⁿ_i=1Aⁱ=1}.