Structured Polynomial Eigenvalue Problems: Good Vibrations from Good Linearizations

Abstract

Many applications give rise to nonlinear eigenvalue problems with an underlying structured matrix polynomial. In this paper several useful classes of structured polynomials (e.g., palindromic, even, odd) are identified and the relationships between them explored. A special class of linearizations which reflect the structure of these polynomials, and therefore preserve symmetries in their spectra, is introduced and investigated. We analyze the existence and uniqueness of such linearizations and show how they may be systematically constructed.

Polynomial Eigenvalue Problems: Theory, Computation, and Structure

“…which, as in the palindromic case detailed before, can be used mutatis mutandis to construct an alternating linearization for P. Similar results were proved for the other flavors of palindromicity and alternation, and concrete examples given in [56].…”

“…This Cayley transformation was extended from matrices to matrix pencils in [50], and in a 1996 paper by Mehrmann [67]. It was then generalized to matrix polynomials in 2006 by Mehrmann and co-authors [56], where it was shown how palindromic and alternating structures are related via a Cayley transformation of matrix polynomials. The definition of general Möbius transformations in [61] completes this development, providing an important and flexible tool for working with matrix polynomials.…”

“…are exactly the Cayley transformations C +1 (P) and C −1 (P), respectively, introduced in [56]. Also note that the reversal operation described in Definition 3 is the Möbius transformation M R corresponding to the matrix…”

“…The pencil spaces L 1 (P) and L 2 (P) introduced by Mehrmann and co-authors [55] were shown in a follow-up paper [56] to provide a rich arena in which to look for linearizations with additional properties like structure preservation or improved numerics, thus realizing the original purpose for their development. Subspaces of pencils that inherit the -palindromic or -alternating structure of P were identified, a constructive method to generate these structured pencils described, and necessary and sufficient conditions for them to be strong linearizations established.…”

“…Now there is a unique pencil in DL(P) corresponding to that vector. This pencil can be explictly constructed using the natural basis for DL(P) mentioned in Section 3.1, and described in detail in [56]. Then reversing the order of the block rows of that DL(P)-pencil turns it into a palindromic pencil in L 1 (P).…”