2019/11/04 by Fernando De Terán, De Terán, Fernando, Andrii Dmytryshyn +3
Computer Science · Mathematics · #15A18 #15A21 #47A56 #65F15 #Advanced Topics in Algebra #FOS: Mathematics #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Rings and Algebras (math.RA) #Tensor decomposition and applications
paper · pdf · doi:10.48550/arxiv.1911.01408
openalex publication_date 2019/11/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We determine the generic complete eigenstructures for n \× n complex\nsymmetric matrix polynomials of odd grade d and rank at most r. More\nprecisely, we show that the set of n \× n complex symmetric matrix\npolynomials of odd grade d, i.e., of degree at most d, and rank at most r\nis the union of the closures of the lfloor rd/2 rfloor+1 sets of symmetric\nmatrix polynomials having certain, explicitly described, complete\neigenstructures. Then, we prove that these sets are open in the set of n\n\× n complex symmetric matrix polynomials of odd grade d and rank at\nmost r. In order to prove the previous results, we need to derive necessary\nand sufficient conditions for the existence of symmetric matrix polynomials\nwith prescribed grade, rank, and complete eigenstructure, in the case where all\ntheir elementary divisors are different from each other and of degree 1. An\nimportant remark on the results of this paper is that the generic\neigenstructures identified in this work are completely different from the ones\nidentified in previous works for unstructured and skew-symmetric matrix\npolynomials with bounded rank and fixed grade larger than one, because the\nsymmetric ones include eigenvalues while the others not. This difference\nrequires to use new techniques.\n