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Generic symmetric matrix polynomials with bounded rank and fixed odd grade. (English) Zbl 1458.15015

Using new techniques, the generic complete eigenstructures for the set \(\mathcal S\) of \(n \times n\) complex symmetric matrix polynomials of odd grade \(d\) and rank at most \(r\) are given. They are completely different from the ones given in previous works for unstructured and skew-symmetric matrix polynomials with bounded rank and fixed grade larger than \(1\), because the symmetric ones include eigenvalues while the others not.
The authors show that \(\mathcal S\) is the union of the closures of the \(\lfloor rd/2\rfloor + 1\) sets of symmetric matrix polynomials having certain, explicitly described, complete eigenstructures, where the topology is induced by a natural Euclidean metric in the space of \(n \times n\) complex symmetric matrices. These bundles are open in \(S\) and the eigenstructures corresponding to these bundles are called generic. In the process of proving the main results, necessary and sufficient conditions are given for the existence of symmetric matrix polynomials with prescribed grade, rank, and complete eigenstructure in the case where all their elementary divisors are different from each other and of degree \(1\).
The paper makes use of matrix theory, topology, and Möbius transformations, and carries a flavor of matrix geometry.
Reviewer: Tin Yau Tam (Reno)

MSC:

15A18 Eigenvalues, singular values, and eigenvectors
15A03 Vector spaces, linear dependence, rank, lineability
15A21 Canonical forms, reductions, classification
15A22 Matrix pencils
47A56 Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones)
65F15 Numerical computation of eigenvalues and eigenvectors of matrices

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mctoolbox; NLEVP
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References:

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