Generalized Standard Triples for Algebraic Linearizations of Matrix Polynomials

We define \emph{generalized standard triples} $\mathbf{X}$, $\mathbf{Y}$, and $L(z) = z\mathbf{C}_{1} - \mathbf{C}_{0}$, where $L(z)$ is a linearization of a regular matrix polynomial $\mathbf{P}(z) \in \mathbb{C}^{n \times n}[z]$, in order to use the representation $\mathbf{X}(z \mathbf{C}_{1}~-~\mathbf{C}_{0})^{-1}\mathbf{Y}~=~\mathbf{P}^{-1}(z)$ which holds except when $z$ is an eigenvalue of $\mathbf{P}$. This representation can be used in constructing so-called \emph{algebraic linearizations} for matrix polynomials of the form $\mathbf{H}(z) = z \mathbf{A}(z)\mathbf{B}(z) + \mathbf{C} \in \mathbb{C}^{n \times n}[z]$ from generalized standard triples of $\mathbf{A}(z)$ and $\mathbf{B}(z)$. This can be done even if $\mathbf{A}(z)$ and $\mathbf{B}(z)$ are expressed in differing polynomial bases. Our main theorem is that $\mathbf{X}$ can be expressed using the coefficients of the expression $1 = \sum_{k=0}^\ell e_k \phi_k(z)$ in terms of the relevant polynomial basis. For convenience, we tabulate generalized 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 common similarity transformations.