Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

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The categories of this paper are: patt-sol
9
The categories of this paper are: nlin
21
The categories of this paper are: adap-org
33
The categories of this paper are: plasm-ph
14
The categories of this paper are: math-ph
26
The categories of this paper are: bayes-an
29
The categories of this paper are: funct-an
17
The categories of this paper are: q-fin
22
The categories of this paper are: chao-dyn
30
The categories of this paper are: q-alg
18
The categories of this paper are: econ
20
The categories of this paper are: acc-phys
28
The categories of this paper are: alg-geom
12
The categories of this paper are: nucl-th
8
The categories of this paper are: hep-ex
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The categories of this paper are: solv-int