The Algebraic Theory of Matrix Polynomials

Abstract

A matrix S is a solvent of the matrix polynomial M(X) Ao Xm +. + Am if M(S) 0 Technical Report [2].If the Ai are scalar matrices, Ai aI, then (1.1) reduces to (1.2) This problem, has been thoroughly studied (Gantmacher,[4]) and we have such classical results as the Cayley-Hamilton theorem and the Lagrange-Sylvester interpolation theorem.

A contour integral approach to the computation of invariant pairs

“…The relation between the Riccati and the quadratic matrix equation is highlighted in [7], whereas a study on the existence of solvents can be found in [13]. Several works address the problem of computing a numerical approximation for the solution of the quadratic matrix equation: an approach to compute, when possible, the dominant solvent is proposed in [12].…”

“…Consider the quadratic matrix solvent problem (see [13], [21]) Note that we cannot construct a solvent whose eigenvalues are 3 and 4 because the associated eigenvectors are linearly dependent. Our approach to compute matrix solvents is based on the relation between the matrix solvent problem (26) and the invariant pair problem (3).…”

Solving a Quadratic Matrix Equation by Newton's Method with Exact Line Searches

“…The standard approach is to reduce (1.2) to a generalized eigenproblem (GEP) Gx = λHx of twice the dimension, 2n. However, as is well known [7], [10], [30], if we can find a solution X of the associated quadratic matrix equation (1.1) then we can write λ 2 A + λB + C = −(B + AX + λA)(X − λI) (1.3) and so the eigenvalues of (1.2) are those of X together with those of the GEP (B + AX)x = −λAx, both of which are n × n problems. Bridges and Morris [5] employ this approach in the solution of differential eigenproblems.…”

“…A solution X of (1.1) is called a solvent [10]. More precisely, X is called a right solvent to distinguish it from a left solvent, which is a solution of X 2 A + XB + C = 0.…”

Asymptotic Stability Analysis of Linear Time- Delay Systems: Delay Dependent Approach

“…In the present paper the majority of presented results start from left solvents of Eigenvalues and eigenvectors of the matrix have a crucial influence on the existence, enumeration and characterization of solvents of the matrix equation (20), (Dennis et al, 1976;Pereira, 2003). Definition 2.2.1 (Dennis et al, 1976;Pereira, 2003).…”