Hermitian matrices with a bounded number of eigenvalues

TL;DR

This paper studies the algebraic structure of Hermitian matrices with limited eigenvalues, providing minimal generating systems, module structures, and extending sum of squares results to complex cases.

Contribution

It introduces new algebraic descriptions and extends known results for Hermitian matrices with bounded eigenvalues, including minimal generating systems and module structures.

Findings

Minimal generating system of the vanishing ideal for 3x3 Hermitian matrices

Structure of the coordinate ring as a module over the special unitary group

Extension of sum of squares results to complex Hermitian matrices

Abstract

Conjugation covariants of matrices are applied to study the real algebraic variety consisting of complex Hermitian matrices with a bounded number of distinct eigenvalues. A minimal generating system of the vanishing ideal of degenerate three by three Hermitian matrices is given, and the structure of the corresponding coordinate ring as a module over the special unitary group is determined. The method applies also for degenerate real symmetric three by three matrices. For arbitrary $n$ partial information on the minimal degree component of the vanishing ideal of the variety of $n\times n$ Hermitian matrices with a bounded number of eigenvalues is obtained, and some known results on sum of squares presentations of subdiscriminants of real symmetric matrices are extended to the case of complex Hermitian matrices.