Mixed discriminants of positive semidefinite matrices

If Ak=(akij), k= 1,2,…,n, are n-by-n matrices, then their mixed discriminant D(A1,…,An) is given by D(A1.....An=1/n! Σσ∈Sa| (αijα(j))| where Sn 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 Dn, which includes the set of n-by-n doubly stochastic matrices, is defined and studied: Dn={(A1.....,An):Ai is a-by-n, positive semidefinite with trace 1, i=1,2,....n; Σni=1Ai=1}