Generic symmetric matrix polynomials with bounded rank and fixed odd grade Articles uri icon

publication date

  • July 2020

start page

  • 1033

end page

  • 1058

issue

  • 3

volume

  • 41

International Standard Serial Number (ISSN)

  • 0895-4798

Electronic International Standard Serial Number (EISSN)

  • 1095-7162

abstract

  • We determine the generic complete eigenstructures for n x n complex symmetric matrix polynomials of odd grade d and rank at most r. More precisely, we show that the set of n x n complex symmetric matrix polynomials of odd grade d , i.e., of degree at most d, and rank at most r is the union of the closures of the rd/2 +1 sets of symmetric matrix polynomials having certain, explicitly described, complete eigenstructures. Then we prove that these sets are open in the set of n x n complex symmetric matrix polynomials of odd grade d and rank at most r. In order to prove the previous results, we need to derive necessary and sufficient conditions 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. An important remark on the results of this paper is that the generic eigenstructures identified in this work are completely different from the ones identified 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. This difference requires using new techniques.

keywords

  • complete eigenstructure; matrix polynomials; symmetry; normal rank; orbits; bundles; genericity; pencils