Palindromic linearizations of a matrix polynomial of odd degreee obtained from Fiedler pencils with repetition

Abstract

Many applications give rise to structured, in particular T- palindromic, matrix polynomials. In order to solve a polynomial eigenvalue problem P(�)x = 0, where P(�) is a T-palindromic matrix polynomial, it is convenient to use palindromic linearizations to ensure that the symmetries in the eigenvalues, elementary divisors, and minimal indices of P(�) due to the palindromicity are preserved. In this paper, new T-palindromic strong linearizations valid for all palindromic matrix polynomials of odd degree are constructed. These linearizations are formulated in terms of Fiedler pencils with repetition, a new family of companion forms that was obtained recently by Antoniou and Vologiannidis.