## Document Type

Presentation Abstract

## Presentation Date

9-30-2019

## Abstract

Given a nonlinear matrix-valued function F(\lambda):\mathbb{C} \longrightarrow \mathbb{C}^{m\times m}, the Non-Linear Eigenvalue Problem (NLEP) consists in computing numbers \lambda \in \mathbb{C} (eigenvalues) and non-zero vectors v \in \mathbb{C}^{m} (eigenvectors) such that

F(\lambda) v = 0,

under the regularity assumption \det(F(z)) \not\equiv 0. NLEPs arise in a variety of applications in Physics and Engineering. Nowadays, a useful approach to tackle them is based on Rational Approximation (RA), which leads to rational eigenvalue problems (REPs). Then, for solving REPs, linearizations of rational matrices are used, which is one of the most competitive methods for this task. In this talk we present the notion of local linearizations of rational matrices. A local linearization of a rational matrix R(\lambda) preserves the zeros and poles of R(\lambda) locally, that is, in subsets of \mathbb{C} and/or at infinity. By using this new notion of linearization, we study the structure of linearizations constructed in the literature for RAs of NLEPs on a target set. Moreover, we provide very simple criteria to determine when a linear polynomial matrix is one of these linearizations.

**Download the attached PDF to see the abstract with proper math formatting.**

## Recommended Citation

Quintana, Maria C., "On the Structure of Linearizations for Rational Approximations of Nonlinear Eigenvalue Problems" (2019). *Colloquia of the Department of Mathematical Sciences*. 582.

https://scholarworks.umt.edu/mathcolloquia/582

## Additional Details

Monday, September 30, 2019 at 3:00 p.m. in Math 103

Refreshments at 4:00 p.m. in Math Lounge 109