Handbook of Linear Algebra, 2nd Edition

Chapter 18

Matrix Polynomials

Jörg Liesen

Technische Universität Berlin

Christian Mehl

Technische Universität Berlin

A matrix polynomial is a polynomial with matrix coefficients. The simplest example comes from the standard linear algebraic eigenvalue problem Ax = λx for a square matrix A. Introducing P(λ) = Iλ1 − Aλ0, which is a matrix polynomial of degree one with the coefficients I and −A, the standard eigenvalue problem can be written in the equivalent form

$P(\lambda )\text{x}=0.$

The generalized eigenvalue problem Ax = λBx (cf. Chapter 56 in this book) can be written in an analogous way with P(λ) = Bλ1 − Aλ0. In order to remind the reader of the strong connection between matrix polynomials and eigenvalue problems, and in agreement with the majority of ...