Standard Triples for Algebraic Linearizations of Matrix Polynomials

Eunice Y. S. Chan, Robert M. Corless, Leili Rafiee Sevyeri

Published 2018-05-11Version 1

Standard triples $\mathbf{X}$, $\mathbf{C}_{1} - \mathbf{C}_{0}$, $\mathbf{Y}$ of nonsingular matrix polynomials $\mathbf{P}(z) \in \mathbb{C}^{r \times r}$ have the property $\mathbf{X}(z \mathbf{C}_{1}~-~\mathbf{C}_{0})^{-1}\mathbf{Y}~=~\mathbf{P}^{-1}(z)$ for $z \notin \Lambda(\mathbf{P}(z))$. They can be used in constructing algebraic linearizations; for example, for $\mathbf{h}(z) = z \mathbf{a}(z)\mathbf{b}(z) + c \in \mathbb{C}^{r \times r}$ from linearizations for $\mathbf{a}(z)$ and $\mathbf{b}(z)$. We tabulate standard triples for orthogonal polynomial bases, the monomial basis, and Newton interpolational bases; for the Bernstein basis; for Lagrange interpolational bases; and for Hermite interpolational bases. We account for the possibility of a transposed linearization, a flipped linearization, and a transposed-and-flipped linearization. We give proofs for the less familiar bases.