The Algebraic Theory of Matrix Polynomials

Abstract

A matrix S is a solvent of the matrix polynomial $M(X) = A_0 X^m + \cdots + A_m $ if $M(S) = 0$ where $A_i ,X,S$ are square matrices. In this paper we develop the algebraic theory of matrix polynomials and solvents. We define division and interpolation, investigate the properties of block Vandermonde matrices, and define and study the existence of a complete set of solvents. We study the relation between the matrix polynomial problem and the lambda-matrix problem, which is to find a scalar $\lambda $ such that $A_0 \lambda ^m + A_1 \lambda ^{m - 1} + \cdots + A_m $ is singular. In a future paper we extend Traub’s algorithm for calculating zeros of scalar polynomials to matrix polynomials and establish global convergence properties of this algorithm for a class of matrix polynomials.

Locations

• SIAM Journal on Numerical Analysis - View
• Columbia Academic Commons (Columbia University) - View - PDF