Hermitian matrices with a bounded number of eigenvalues

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. MSC: Primary: 13F20, 13A50, 14P05; Secondary: 15A15, 15A72, 20G05, 22E47

Real Symmetric Matrices with Partitioned Eigenvalues

“…Matrices with repeated eigenvalues have been studied in contexts from geometry [3] to classical invariant theory and linear algebra [6,7,8,14,16,18,19,20]. Recently, they have come to focus in the study of curvature of algebraic varieties [4].…”

“…In a refinement of the study of degenerate matrices, some authors [7,8,19] have studied matrices by their number of distinct eigenvalues. In this situation, the role of the matrix discriminant is played by the set of k-subdiscriminants.…”

HIGHEST-WEIGHT VECTORS FOR THE ADJOINT ACTION OF GL n ON POLYNOMIALS, II

“…So the stable multiplicity (see [6] for more about this) of L C (χ t ) in the degree d-piece of k[N ] is the number of partitions of d of length t. For the weight χ t changing the order of the arguments in (8) gives the same element up to a sign. So (8) gives us a candidate spanning set labelled by partitions of length t (lower all indices by 1) and by the above the elements of this set labelled by partitions of degree d and length t would have to be independent for big n. 5. Computer calculations show that the answer to the question from [24, §4] (Remark 3 above) is in general no.…”

“…Computer calculations show that the answer to the question from [24, §4] (Remark 3 above) is in general no. For example, for n = 4 and χ = (2, 1, −1, −2) the elements from (8) give only a 1-dimensional space of invariants in degree 6 and not the required 2 dimensions. However, it turns out that in all cases were I found that the elements from (8) don't span, replacing E λ by a suitable Sym t × Sym t -conjugate does give a spanning set.…”