Mixed discriminant - Knowlpedia

Definition

Let $A_1,\ldots,A_n$ be $n\times n$ matrices in a class on which a degree- $n$ determinant is defined. Their mixed discriminant is

D(A_1,\ldots,A_n) =\frac1{n!}[t_1\cdots t_n]\, \det(t_1A_1+\cdots+t_nA_n),

where $[t_1\cdots t_n]$ denotes the coefficient of that monomial. This normalization gives $D(A,\ldots,A)=\det A$ .

Properties

The mixed discriminant is symmetric and multilinear. For real symmetric or complex Hermitian matrices it polarizes the ordinary determinant; for hyperhermitian matrices it polarizes the Moore determinant. It is nonnegative when all its arguments are positive semidefinite.

Role in Hessian measures

Applying $D$ to Hessians of several functions produces mixed Monge–Ampère expressions. In quaternionic pluripotential theory this leads to the mixed quaternionic Monge–Ampère measure. This matrix construction is analogous to, but distinct from, the mixed volume of convex bodies.

References

A. D. Aleksandrov, “Mixed discriminants and mixed volumes,” Matematicheskii Sbornik 3 (1938), 227–251.
Semyon Alesker, “Quaternionic plurisubharmonic functions and their applications to convexity,” 2016 revision. arXiv record. Relevant: §1.