Mixed discriminants of positive semidefinite matrices
If \(A^ k=(a^ k_{ij})\) are \(n\times n\) complex matrices \(k=1,2,...,n\), then their mixed discriminant \(D(A^ 1,...,A^ n)\) is \(\frac{1}{n!}\sum_{\sigma \in S_ n}\det (a_{ij}^{\sigma (j)})\), where \(S_ n\) is the symmetric group of degree n. If all the \(A^ k\) are equal this turns out to be det A, whereas if each \(A^ k\) is a diagonal matrix the mixed discriminant equals \(\frac{1}{n!}per(a^ j_{ii})\). The author gives several alternative ways of defining \(D(A^ 1,...,A^ n)\) and he gives a simple proof of a result of \textit{J. S. Lomont} and \textit{M. S. Cheema} [Linear, Multilinear Algebra, 14, 199-223 (1979; Zbl 0521.15018)] which gives for mixed discriminants an analogue of Ryser's formula for the permanent. He expresses the mixed discriminant as an inner product and derives a Cauchy-Binet formula. He also proves, for nonnegative definite Hermitian matrices, a generalization of König's theorem on 0-1 matrices. He proves that if \(A^ 1,...,A^ n\) are \(n\times n\) nonnegative definite Hermitian matrices, each having trace 1, with \(A^ 1+...+A^ n=I\) then \(D(A^ 1,...,A^ n)>0\) and he gives a partial characterization of the extreme points of the set of n-tuples \((A^ 1,...,A^ n)\) with these properties, in an attempt to get an analogue of the Birkhoff-von Neumann theorem for doubly stochastic matrices.
scientific article; zbMATH DE number 4139938
scientific article