Local Linearizations of Rational Matrices with Application to Rational Approximations of Nonlinear Eigenvalue Problems

Abstract

This paper presents a definition for local linearizations of rational matrices and studies their properties. This definition allows us to introduce matrix pencils associated to a rational matrix that preserve its structure of zeros and poles in subsets of any algebraically closed field and also at infinity. Moreover, such definition includes, as particular cases, other definitions that have been used previously in the literature. In this way, this new theory of local linearizations captures and explains rigorously the properties of all the different pencils that have been used from the 1970's until 2019 for computing zeros, poles and eigenvalues of rational matrices. Particular attention is paid to those pencils that have appeared recently in the numerical solution of nonlinear eigenvalue problems through rational approximation.

Linearizations of rational matrices from general representations

“…which shows that T r maps vectors in the right nullspace of R(λ) to vectors in the right nullspace of P (λ). Finally, by (7) and the rank-nullity theorem, we have…”

“…have no eigenvalues in F. In such case, we also say that the realization in ( 5) is minimal. If the matrices in (8) have no eigenvalues in a subset Ω ⊆ F then P (λ) is said to be minimal in Ω [7,Definition 3.3]. In such case, we also say that the realization ( 5) is minimal in Ω.…”

“…In such case, we also say that the realization ( 5) is minimal in Ω. The main property of having minimality is that if the polynomial system matrix P (λ) is minimal in Ω, then the poles of R(λ) in Ω are the zeros of the state matrix A(λ) in Ω, and the zeros of R(λ) in Ω are the zeros of P (λ) in Ω (see [7,Theorem 3.7]). Poles and zeros of a rational matrix are defined through the notion of the Smith-McMillan form [16], other more recent references for the Smith-McMillan form of a rational matrix are [14,17,23].…”

“…We now recall the notion of (strong) block minimal basis pencil. It will be our main tool for building linearizations of rational matrices in the sense of [7].…”

“…A competitive method for solving REPs is linearization [1,2,7,8,19]. Linearization transforms the REP into a generalized eigenvalue problem in such a way that the pole and zero information of the corresponding rational matrix is preserved.…”