Hermitian matrices with a bounded number of eigenvalues

Domokos, Mátyás (2013) Hermitian matrices with a bounded number of eigenvalues. LINEAR ALGEBRA AND ITS APPLICATIONS, 439 (12). pp. 3964-3979. ISSN 0024-3795

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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 × 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.

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