Companion matrices of matrix polynomials L(λ) (with possibly singular leading coefficient) are a familiar tool in matrix theory and numerical practice leading to so-called "linearizations" λB − A of the polynomials. Matrix polynomials as approximations to more general matrix functions lead to the study of matrix polynomials represented in a variety of classical systems of polynomials, including orthogonal systems and Lagrange polynomials, for example. For several such representations, it is shown how to construct (strong) linearizations via analogous companion matrix pencils. In case L(λ) has Hermitian or alternatively complex symmetric coefficients, the determination of linearizations λB − A with A and B Hermitian or complex symmetric is also discussed.

We collect here elementary properties of differentiation matrices for univariate polynomials expressed in various bases, including orthogonal polynomial bases and non-degree-graded bases such as Bernstein bases and Lagrange & Hermite interpolational bases. Alternatively, we might work with f (x) = b T φ φ φ T (x) and in that case use the transpose of D, in b T = a T D T .

A classical Rayleigh-quotient iterative algorithm (known as "broken iteration") for finding eigenvalues and eigenvectors is applied to semisimple regular matrix pencils A − λB. It is proved that cubic convergence is attained for eigenvalues and superlinear convergence of order three for eigenvectors. Also, each eigenvalue has a local basin of attraction. A closely related Newton algorithm is examined. Numerical examples are included.

Explicit differentiation matrices in various polynomial bases are presented in this work. The idea is to avoid any change of basis in the process of polynomial differentiation. This article concerns both degree-graded polynomial bases such as orthogonal bases, and non-degree-graded polynomial bases including the Lagrange and Bernstein bases.