Standard triples of structured matrix polynomials

Al-Ammari, Maha; Tisseur, Françoise

doi:10.1016/j.laa.2012.03.020

Cited by 14 publications

(16 citation statements)

References 15 publications

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“…) is a standard triple for P (z). This is also reported in Lemma 2 in [1]. There are two other representations of a matrix polynomial given a standard triple: the right canonical form, and the left canonical form.…”

Section: Introductionmentioning

confidence: 56%

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Generalized Standard Triples for Algebraic Linearizations of Matrix Polynomials

Chan

Corless

Sevyeri

2021

ELA

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We define generalized standard triples $\boldsymbol{X}$, $\boldsymbol{Y}$, and $L(z) = z\boldsymbol{C}_{1} - \boldsymbol{C}_{0}$, where $L(z)$ is a linearization of a regular matrix polynomial $\boldsymbol{P}(z) \in \mathbb{C}^{n \times n}[z]$, in order to use the representation $\boldsymbol{X}(z \boldsymbol{C}_{1}~-~\boldsymbol{C}_{0})^{-1}\boldsymbol{Y}~=~\boldsymbol{P}^{-1}(z)$ which holds except when $z$ is an eigenvalue of $\boldsymbol{P}$. This representation can be used in constructing so-called algebraic linearizations for matrix polynomials of the form $\boldsymbol{H}(z) = z \boldsymbol{A}(z)\boldsymbol{B}(z) + \boldsymbol{C} \in \mathbb{C}^{n \times n}[z]$ from generalized standard triples of $\boldsymbol{A}(z)$ and $\boldsymbol{B}(z)$. This can be done even if $\boldsymbol{A}(z)$ and $\boldsymbol{B}(z)$ are expressed in differing polynomial bases. Our main theorem is that $\boldsymbol{X}$ 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.

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Section: Introductionmentioning

confidence: 56%

“…Some other recent papers of interest include [4,28,12,17], and [32]; this is a very active area. See also [1]. In that paper, standard triples for structured matrices are studied.…”

Section: Introductionmentioning

confidence: 99%

Generalized Standard Triples for Algebraic Linearizations of Matrix Polynomials

Chan

Corless

Sevyeri

2021

ELA

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“…Let P(x) be the matrix polynomial defined in (1). We give here a brief overview of those aspects in the spectral theory of square complex matrix polynomials that are relevant to this paper.…”

Section: Matrix Polynomial Theorymentioning

confidence: 99%

The Limit Empirical Spectral Distribution of Gaussian Monic Complex Matrix Polynomials

Barbarino

Noferini

2022

J Theor Probab

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We define the empirical spectral distribution (ESD) of a random matrix polynomial with invertible leading coefficient, and we study it for complex $$n \times n$$ n × n Gaussian monic matrix polynomials of degree k. We obtain exact formulae for the almost sure limit of the ESD in two distinct scenarios: (1) $$n \rightarrow \infty $$ n → ∞ with k constant and (2) $$k \rightarrow \infty $$ k → ∞ with n constant. The main tool for our approach is the replacement principle by Tao, Vu and Krishnapur. Along the way, we also develop some auxiliary results of potential independent interest: We slightly extend a result by Bürgisser and Cucker on the tail bound for the norm of the pseudoinverse of a nonzero mean matrix, and we obtain several estimates on the singular values of certain structured random matrices.

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“…[ 6 – 13 ]). The spectral method basically uses the construction of standard pairs and standard triples (e.g., [ 14 , 15 ]) to solve differential equations.…”

Section: Applicationsmentioning

confidence: 99%