Matrix equations. Factorization of matrix polynomials

Abstract

This chapter discusses the main results and methods in the theory of matrix equations, with emphasis on algebraic aspects of the theory of matrix equations and matrix polynomials. The problem of factorization of matrix polynomials is essentially a problem concerning special systems of matrix equations. The invariant subspaces of certain associated matrices and linear matrix pencils are used, thus reducing the problem of solving matrix equations to certain spectral matrix problems. Linear and nonlinear matrix equations are presented in the chapter. The chapter describes the approaches based on perturbation theory and iterative methods. All scalars, vectors, and matrices described are complex. A regular linear matrix pencil is a λ-matrix λ B – A with N × N matrices A and B , provided that the scalar polynomial p (λ) = det(λ B – A ) is not identically zero. If the degree of the polynomial p (λ) is less than N , then the pencil λ B – A is said to have an infinite eigenvalue. Monic and comonic block eigenpairs of a pencil λ B – A are, evidently, particular cases of decomposable block eigenpairs.