A Geometric Description of the Sets of Palindromic and Alternating Matrix Pencils with Bounded Rank Articles uri icon

publication date

  • July 2018

start page

  • 1116

end page

  • 1134

issue

  • 3

volume

  • 39

International Standard Serial Number (ISSN)

  • 0895-4798

Electronic International Standard Serial Number (EISSN)

  • 1095-7162

abstract

  • The sets of n x n T-palindromic, T-antipalindromic, T-even, and T-odd matrix pencils with rank at most r < n are algebraic subsets of the set of n x n matrix pencils. In this paper, we determine their dimension and we prove that they are all irreducible. This is in contrast with the nonstructured case, since it is known that the set of n x matrix pencils with rank at most r< n is an algebraic set with r + 1 irreducible components. We also show that these sets of structured pencils with bounded rank are the closure of the congruence orbit of a certain structured pencil given in canonical form. This allows us to determine the generic canonical form of a structured n x n matrix pencil with rank at most r, for any of the previous structures.

keywords

  • matrix pencil; t-palindromic; t-alternating; strict equivalence; congruence; orbit; spectral information; algebraic set