Spectral equivalence of matrix polynomials and the Index Sum Theorem Articles uri icon

publication date

  • October 2014

start page

  • 264

end page

  • 333

volume

  • 459

International Standard Serial Number (ISSN)

  • 0024-3795

Electronic International Standard Serial Number (EISSN)

  • 1873-1856

abstract

  • The concept of linearization is fundamental for theory, applications, and spectral computations related to matrix polynomials. However, recent research on several important classes of structured matrix polynomials arising in applications has revealed that the strategy of using linearizations to develop structure-preserving numerical algorithms that compute the eigenvalues of structured matrix polynomials can be too restrictive, because some structured polynomials do not have any linearization with the same structure. This phenomenon strongly suggests that linearizations should sometimes be replaced by other low degree matrix polynomials in applied numerical computations. Motivated by this fact, we introduce equivalence relations that allow the possibility of matrix polynomials (with coefficients in an arbitrary field) to be equivalent, with the same spectral structure, but have different sizes and degrees. These equivalence relations are directly modeled on the notion of linearization, and consequently inherit the simplicity, applicability, and most relevant properties of linearizations; simultaneously, though, they are much more flexible in the possible degrees of equivalent polynomials. This flexibility allows us to define in a unified way the notions of quadratification and ℓ-ification, to introduce the concept of companion form of arbitrary degree, and to provide concrete and simple examples of these notions that generalize in a natural and smooth way the classical first and second Frobenius companion forms. [...]

keywords

  • algorithms; eigenvalues and eigenfunctions; indexing (materials working); linearization; polynomials; set theory; companion form; elementary divisors; index sum theorem; matrix pencil; matrix polynomials; minimal indices; partial multiplicity sequence; quadratification; regular; singular; spectral equivalence; structural indices; structured matrixes; unimodular; matrix algebra