#### Generalized Standard Triples for Algebraic Linearizations of Matrix Polynomials

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

We define "generalized standard triples" $\mathbf{X}$, $z\mathbf{C}_{1} - \mathbf{C}_{0}$, $\mathbf{Y}$ of regular matrix polynomials $\mathbf{P}(z) \in \mathbb{C}^{n \times n}$ in order to use the representation $\mathbf{X}(z \mathbf{C}_{1} - \mathbf{C}_{0})^{-1}\mathbf{Y} = \mathbf{P}^{-1}(z)$ for $z \notin \Lambda(\mathbf{P}(z))$. This representation can be used in constructing algebraic linearizations; for example, for $\mathbf{H}(z) = z \mathbf{A}(z)\mathbf{B}(z) + \mathbf{C} \in \mathbb{C}^{n \times n}$ from linearizations for $\mathbf{A}(z)$ 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. We give explicit proofs for the less familiar bases.

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