Summary: PALINDROMIC LINEARIZATIONS OF A MATRIX POLYNOMIAL OF ODD
DEGREE OBTAINED FROM FIEDLER PENCILS WITH REPETITION.
M.I. BUENO AND S. FURTADO
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, we construct new palindromic
strong linearizations valid for all palindromic matrix polynomials of odd degree. These linearizations are formulated
in terms of Fiedler pencils with repetition, a new family of companion forms that was obtained recently by Antoniou
Key words. matrix polynomials, linearization, Fiedler pencils with repetition, T-palindromic linearizations,
companion form, polynomial eigenvalue problem.
AMS subject classifications. 65F15, 15A18, 15A22.
1. Introduction. Let F be a field and denote by Mn(F) the set of n × n matrices over
P() = Akk
+ · · · + A0, (1.1)
where Ai Mn(F), be a matrix polynomial of degree k 2 (that is, Ak = 0). The matrix
polynomial P() is said to be regular if det(P()) 0. Otherwise, P() is said to be