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## Relative asymptotics for orthogonal matrix polynomials

Fonte: Centro de Matemática da Universidade de Coimbra
Publicador: Centro de Matemática da Universidade de Coimbra

Tipo: Pré-impressão

ENG

Relevância na Pesquisa

56.42%

#Matrix orthogonal polynomials#Linear functional#Recurrence relation#Tridiagonal operator#Asymptotic results#Nevai class

In this paper we study sequences of matrix polynomials that satisfy a
non-symmetric recurrence relation. To study this kind of sequences we use a vector
interpretation of the matrix orthogonality. In the context of these sequences of
matrix polynomials we introduce the concept of the generalized matrix Nevai class
and we give the ratio asymptotics between two consecutive polynomials belonging to
this class. We study the generalized matrix Chebyshev polynomials and we deduce
its explicit expression as well as we show some illustrative examples. The concept of
a Dirac delta functional is introduced. We show how the vector model that includes
a Dirac delta functional is a representation of a discrete Sobolev inner product. It
also allows to reinterpret such perturbations in the usual matrix Nevai class. Finally,
the relative asymptotics between a polynomial in the generalized matrix Nevai class
and a polynomial that is orthogonal to a modification of the corresponding matrix
measure by the addition of a Dirac delta functional is deduced.

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## Relative asymptotics for orthogonal matrix polynomials with respect to a perturbed matrix measure on the unit circle

Fonte: Springer
Publicador: Springer

Tipo: Artigo de Revista Científica
Formato: application/pdf

Publicado em /12/2002
ENG

Relevância na Pesquisa

56.28%

Given a positive definite matrix measure Ω supported on the unit circle T, then main purpose of this paper is to study the asymptotic behavior of $L_n( ilde{Omega}) L_n(Omega) -1}$ and $Phi_n(z, ilde{Omega}) Phi_n(z, ilde{Omega}) -1}$ where $ ilde{Omega}(z) = Omega(z) + M delta ( z - w)$, $ 1$, M is a positive definite matrix and δ is the Dirac matrix measure. Here, Ln(·) means the leading coefficient of the orthonormal matrix polynomials Φn(z; •).; Finally, we deduce the asymptotic behavior of $Phi_n(omega, ilde{Omega}) Phi_n(omega, Omega)$ in the case when M=I.; The work of the second author was supported by Dirección General de Enseñanza Superior (DGES) of Spain under grant PB96-0120-C03-01 and INTAS
Project INTAS93-0219 Ext.; 19 pages, no figures.-- MSC2000 codes: 42C05, 47A56.; MR#: MR1970413 (2004b:42058); Zbl#: Zbl 1047.42021

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## Perturbations in the Nevai matrix class of orthogonal matrix polynomials

Fonte: Elsevier
Publicador: Elsevier

Tipo: Artigo de Revista Científica

Publicado em 15/10/2001
ENG

Relevância na Pesquisa

56.19%

In this paper we study a Jacobi block matrix and the behavior of the limit of its entries when a perturbation of its spectral matrix measure by the addition of a Dirac delta matrix measure is introduced.; The work of the second author was supported by Dirección General de Enseñanza Superior (DGES) of Spain under grant PB96-0120-CO3-01 and INTAS Project INTAS-93-0219 Ext, and the work of the third author was supported by DGES under grant PB 95-1205, INTAS-93-0219-ext and Junta de Andalucía, Grupo de Investigación FQM 229.; 24 pages, no figures.-- MSC2000 codes: 15A54, 15A21, 42C05.; MR#: MR1855403 (2002i:42037); Zbl#: Zbl 0992.15022

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## Deflation for block eigenvalues of block partitioned matrices with an application to matrix polynomials of commuting matrices

Fonte: Universidade de Coimbra
Publicador: Universidade de Coimbra

Tipo: Artigo de Revista Científica
Formato: aplication/PDF

ENG

Relevância na Pesquisa

66.22%

A method for computing a complete set of block eigenvalues for a block partitioned matrix using a generalized form of Wielandt's deflation is presented. An application of this process is given to compute a complete set of solvents of matrix polynomials where the coefficients and the variable are commuting matrices.; http://www.sciencedirect.com/science/article/B6TYJ-444G6J6-H/1/c3a6cac92904c18a24a2d28ebf602b0f

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## Matrix polynomials with partially prescribed eigenstructure: eigenvalue sensitivity and condition estimation

Fonte: Sociedade Brasileira de Matemática Aplicada e Computacional
Publicador: Sociedade Brasileira de Matemática Aplicada e Computacional

Tipo: Artigo de Revista Científica
Formato: text/html

Publicado em 01/12/2005
EN

Relevância na Pesquisa

56.25%

#matrix polynomials#block companion matrices#departure from normality#eigenvalue sensitivity#controllability Gramians

Let Pm(z) be a matrix polynomial of degree m whose coefficients At Î Cq×q satisfy a recurrence relation of the form: h kA0+ h k+1A1+...+ h k+m-1Am-1 = h k+m, k > 0, where h k = RZkL Î Cp×q, R Î Cp×n, Z = diag (z1,...,z n) with z i ¹ z j for i ¹ j, 0 < |z j| < 1, and L Î Cn×q. The coefficients are not uniquely determined from the recurrence relation but the polynomials are always guaranteed to have n fixed eigenpairs, {z j,l j}, where l j is the jth column of L*. In this paper, we show that the z j's are also the n eigenvalues of an n×n matrix C A; based on this result the sensitivity of the z j's is investigated and bounds for their condition numbers are provided. The main result is that the z j's become relatively insensitive to perturbations in C A provided that the polynomial degree is large enough, the number n is small, and the eigenvalues are close to the unit circle but not extremely close to each other. Numerical results corresponding to a matrix polynomial arising from an application in system theory show that low sensitivity is possible even if the spectrum presents clustered eigenvalues.

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## Matrix polynomials with completely prescribed eigenstructure

Fonte: Society for Industrial and Applied Mathematics
Publicador: Society for Industrial and Applied Mathematics

Tipo: info:eu-repo/semantics/publishedVersion; info:eu-repo/semantics/article; info:eu-repo/semantics/conferenceObject

Publicado em /07/2014
ENG

Relevância na Pesquisa

66.32%

#Matrix polynomials#Index sum theorem#Invariant polynomials#(l)-cations#Minimal indices#Inverse polynomial eigenvalue problems.#Matemáticas

We present necessary and su cient conditions for the existence of a matrix polynomial when its degree, its nite and in nite elementary divisors, and its left and right minimal indices are prescribed. These conditions hold for arbitrary in nite elds and are determined mainly by the \index sum theorem", which is a fundamental relationship between the rank, the degree, the sum of all partial multiplicities, and the sum of all minimal indices of any matrix polynomial. The proof developed for the existence of such polynomial is constructive and, therefore, solves a very general inverse problem for matrix polynomials with prescribed complete eigenstructure. This result allows us to x the problem of the existence of (l)-ifications of a given matrix polynomial, as well as to determine all their possible sizes and eigenstructures.; This research was partially supported by the Ministerio de Economía y Competitividad of Spain through grant MTM-2012-32542 and by the Belgian Network DYSCO (Dynamical Systems, Control, and Optimization), funded by the Interuniversity Attraction Poles Programme, initiated by the Belgian State, Science Policy Office.; The proceeding at: Joint ALAMA-GAMM/ANLA 2014 Meeting, took place 2014, July 14-16, in Barcelona (Spain).

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## Spectral equivalence of matrix polynomials and the Index Sum Theorem

Fonte: Elsevier
Publicador: Elsevier

Tipo: info:eu-repo/semantics/acceptedVersion; info:eu-repo/semantics/article

Publicado em /10/2014
ENG

Relevância na Pesquisa

46.22%

#Algorithms#Eigenvalues and eigenfunctions#Indexing (materials working)#Linearization#Polynomials#Set theory#Companion form#Elementary divisors#Index Sum Theorem#Matrix pencil#Matrix polynomials

This research was partially supported by Ministerio de Ciencia e Innovación of Spain through grant MTM-2009-09281, and by Ministerio de Economía y Competitividad of Spain through grant MTM2012-32542.

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## Large vector spaces of block-symmetric strong linearizations of matrix polynomials

Fonte: Elsevier
Publicador: Elsevier

Tipo: info:eu-repo/semantics/acceptedVersion; info:eu-repo/semantics/article

Publicado em 15/07/2015
ENG

Relevância na Pesquisa

56.22%

#block-symmetric linearizations#Fiedler pencils with repetition#generalized Fiedler pencils with repetition#matrix polynomials#strong linearizations#structured matrix polynomials#vector space DL(P)#Matemáticas

Given a matrix polynomial P(lambda) = Sigma(k)(i=0) lambda(i) A(i) of degree k, where A(i) are n x n matrices with entries in a field F, the development of linearizations of P(lambda) that preserve whatever structure P(lambda) might posses has been a very active area of research in the last decade. Most of the structure-preserving linearizations of P(lambda) discovered so far are based on certain modifications of block-symmetric linearizations. The block-symmetric linearizations of P(lambda) available in the literature fall essentially into two classes: linearizations based on the so-called Fiedler pencils with repetition, which form a finite family, and a vector space of dimension k of block-symmetric pencils, called DL(P), such that most of its pencils are linearizations. One drawback of the pencils in DL(P) is that none of them is a linearization when P(lambda) is singular. In this paper we introduce new vector spaces of block,symmetric pencils, most of which are strong linearizations of P(lambda). The dimensions of these spaces are O(n(2)), which, for n >= root k, are much larger than the dimension of DL(P). When k is odd, many of these vector spaces contain linearizations also when P(lambda) is singular. The coefficients of the block-symmetric pencils in these new spaces can be easily constructed as k x k block-matrices whose n x n blocks are of the form 0...

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## Relative asymptotics for orthogonal matrix polynomials

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 03/03/2010

Relevância na Pesquisa

46.39%

In this paper we study sequences of matrix polynomials that satisfy a
non-symmetric recurrence relation. To study this kind of sequences we use a
vector interpretation of the matrix orthogonality. In the context of these
sequences of matrix polynomials we introduce the concept of the generalized
matrix Nevai class and we give the ratio asymptotics between two consecutive
polynomials belonging to this class. We study the generalized matrix Chebyshev
polynomials and we deduce its explicit expression as well as we show some
illustrative examples. The concept of a Dirac delta functional is introduced.
We show how the vector model that includes a Dirac delta functional is a
representation of a discrete Sobolev inner product. It also allows to
reinterpret such perturbations in the usual matrix Nevai class. Finally, the
relative asymptotics between a polynomial in the generalized matrix Nevai class
and a polynomial that is orthogonal to a modification of the corresponding
matrix measure by the addition of a Dirac delta functional is deduced.

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## Orthogonal matrix polynomials satisfying differential equations with recurrence coefficients having non-scalar limits

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 08/02/2011

Relevância na Pesquisa

46.31%

We introduce a family of weight matrices $W$ of the form $T(t)T^*(t)$,
$T(t)=e^{\mathscr{A}t}e^{\mathscr{D}t^2}$, where $\mathscr{A}$ is certain
nilpotent matrix and $\mathscr{D}$ is a diagonal matrix with negative real
entries. The weight matrices $W$ have arbitrary size $N\times N$ and depend on
$N$ parameters.
The orthogonal polynomials with respect to this family of weight matrices
satisfy a second order differential equation with differential coefficients
that are matrix polynomials $F_2$, $F_1$ and $F_0$ (independent of $n$) of
degrees not bigger than 2, 1 and 0 respectively.
For size $2\times 2$, we find an explicit expression for a sequence of
orthonormal polynomials with respect to $W$. In particular, we show that one of
the recurrence coefficients for this sequence of orthonormal polynomials does
not asymptotically behave as a scalar multiple of the identity, as it happens
in the examples studied up to now in the literature.; Comment: 17 pages

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## Pure states, positive matrix polynomials and sums of hermitian squares

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

46.3%

#Mathematics - Operator Algebras#Mathematics - Optimization and Control#15A48, 11E25, 13J30 (Primary), 15A54, 14P10, 46A55 (Secondary)

Let M be an archimedean quadratic module of real t-by-t matrix polynomials in
n variables, and let S be the set of all real n-tuples where each element of M
is positive semidefinite. Our key finding is a natural bijection between the
set of pure states of M and the cartesian product of S with the real projective
(t-1)-space. This leads us to conceptual proofs of positivity certificates for
matrix polynomials, including the recent seminal result of Hol and Scherer: If
a symmetric matrix polynomial is positive definite on S, then it belongs to M.
We also discuss what happens for non-symmetric matrix polynomials or in the
absence of the archimedean assumption, and review some of the related classical
results. The methods employed are both algebraic and functional analytic.; Comment: 15 pages, accepted for publication in Indiana Univ. Math. J

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## On the distance from a weakly normal matrix polynomial to matrix polynomials with a prescribed multiple eigenvalue

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 12/10/2014

Relevância na Pesquisa

46.25%

Consider an $n \times n$ matrix polynomial $P(\lambda)$. An upper bound for a
spectral norm distance from $P(\lambda)$ to the set of $n \times n$ matrix
polynomials that have a given scalar $\mu\in\mathbb{C}$ as a multiple
eigenvalue was recently obtained by Papathanasiou and Psarrakos (2008). This
paper concerns a refinement of this result for the case of weakly normal matrix
polynomials. A modification method is implemented and its efficiency is
verified by an illustrative example.

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## Orthogonal matrix polynomials and higher order recurrence relations

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 04/10/1993

Relevância na Pesquisa

46.28%

It is well-known that orthogonal polynomials on the real line satisfy a
three-term recurrence relation and conversely every system of polynomials
satisfying a three-term recurrence relation is orthogonal with respect to some
positive Borel measure on the real line. In this paper we extend this result
and show that every system of polynomials satisfying some $(2N+1)$-term
recurrence relation can be expressed in terms of orthonormal matrix
polynomials for which the coefficients are $N\times N$ matrices. We apply this
result to polynomials orthogonal with respect to a discrete Sobolev inner
product and other inner products in the linear space of polynomials. As an
application we give a short proof of Krein's characterization of orthogonal
polynomials with a spectrum having a finite number of accumulation points.

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## Backward errors and linearizations for palindromic matrix polynomials

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

46.18%

We derive computable expressions of structured backward errors of approximate
eigenelements of *-palindromic and *-anti-palindromic matrix polynomials. We
also characterize minimal structured perturbations such that approximate
eigenelements are exact eigenelements of the perturbed polynomials. We detect
structure preserving linearizations which have almost no adverse effect on the
structured backward errors of approximate eigenelements of the *-palindromic
and *-anti-palindromic polynomials.; Comment: 19 pages, submitted

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## On Pseudospectra of Matrix Polynomials and their Boundaries

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

46.18%

In the first part of this paper, the main concern is with smoothness
properties of the boundary of the pseudospectrum of a matrix polynomial. In the
second part, results are obtained concerning the number of connected components
of pseudospectra, as well as results concerning matrix polynomials with
multiple eigenvalues, or the proximity to such polynomials.; Comment: 31 pages, 5 figures, recently submitted to Mathematics of
Computation. In this revised version we have corrected some further typos and
added an extra example

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## Matrix polynomials, generalized Jacobians, and graphical zonotopes

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 16/06/2015

Relevância na Pesquisa

46.32%

A matrix polynomial is a polynomial in a complex variable $\lambda$ with
coefficients in $n \times n$ complex matrices. The spectral curve of a matrix
polynomial $P(\lambda)$ is the curve $\{ (\lambda, \mu) \in \mathbb{C}^2 \mid
\mathrm{det}(P(\lambda) - \mu \cdot \mathrm{Id}) = 0\}$. The set of matrix
polynomials with a given spectral curve $C$ is known to be closely related to
the Jacobian of $C$, provided that $C$ is smooth. We extend this result to the
case when $C$ is an arbitrary nodal, possibly reducible, curve. In the latter
case the set of matrix polynomials with spectral curve $C$ turns out to be
naturally stratified into smooth pieces, each one being an open subset in a
certain generalized Jacobian. We give a description of this stratification in
terms of purely combinatorial data and describe the adjacency of strata. We
also make a conjecture on the relation between completely reducible matrix
polynomials and the canonical compactified Jacobian defined by V.Alexeev.; Comment: 19 pages, 7 figures

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## On the distance from a matrix polynomial to matrix polynomials with two prescribed eigenvalues

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

46.26%

Consider an $n \times n$ matrix polynomial $P(\lambda)$. A spectral norm
distance from $P(\lambda)$ to the set of $n \times n$ matrix polynomials that
have a given scalar $\mu\in\mathbb{C}$ as a multiple eigenvalue was introduced
and obtained by Papathanasiou and Psarrakos. They computed lower and upper
bounds for this distance, constructing an associated perturbation of
$P(\lambda)$. In this paper, we extend this result to the case of two given
distinct complex numbers $\mu_{1}$ and $\mu_{2}$. First, we compute a lower
bound for the spectral norm distance from $P(\lambda)$ to the set of matrix
polynomials that have $\mu_1,\mu_2$ as two eigenvalues. Then we construct an
associated perturbation of $P(\lambda)$, such that the perturbed matrix
polynomial has two given scalars $\mu_1$ and $\mu_2$ in its spectrum. Finally,
we derive an upper bound for the distance by the constructed perturbation of
$P(\lambda)$. Numerical examples are provided to illustrate the validity of the
method.

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## Second order differential operators having several families of orthogonal matrix polynomials as eigenfunctions

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

46.34%

The aim of this paper is to bring into the picture a new phenomenon in the
theory of orthogonal matrix polynomials satisfying second order differential
equations. The last few years have witnessed some examples of a (fixed) family
of orthogonal matrix polynomials whose elements are common eigenfunctions of
several linearly independent second order differential operators. We show that
the dual situation is also possible: there are examples of different families
of matrix polynomials, each family orthogonal with respect to a different
weight matrix, whose elements are eigenfunctions of a common second order
differential operator.
These examples are constructed by adding a discrete mass at certain point to
a weight matrix: $\widetilde{W}=W+\delta_{t_0}M(t_0)$. Our method consists in
showing how to choose the discrete mass $M(t_0)$, the point $t_0$ where the
mass lives and the weight matrix $W$ so that the new weight matrix
$\widetilde{W}$ inherits some of the symmetric second order differential
operators associated with $W$. It is well known that this situation is not
possible for the classical scalar families of Hermite, Laguerre and Jacobi.; Comment: 16 pages

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## Locating the eigenvalues of matrix polynomials

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

46.24%

Some known results for locating the roots of polynomials are extended to the
case of matrix polynomials. In particular, a theorem by A.E. Pellet [Bulletin
des Sciences Math\'ematiques, (2), vol 5 (1881), pp.393-395], some results of
D.A. Bini [Numer. Algorithms 13:179-200, 1996] based on the Newton polygon
technique, and recent results of M. Akian, S. Gaubert and M. Sharify (see in
particular [LNCIS, 389, Springer p.p.291-303] and [M. Sharify, Ph.D. thesis,
\'Ecole Polytechnique, ParisTech, 2011]). These extensions are applied for
determining effective initial approximations for the numerical computation of
the eigenvalues of matrix polynomials by means of simultaneous iterations, like
the Ehrlich-Aberth method. Numerical experiments that show the computational
advantage of these results are presented.; Comment: Submitted to SIMAX

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## Spectral Sparsification of Random-Walk Matrix Polynomials

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 11/02/2015

Relevância na Pesquisa

46.31%

#Computer Science - Data Structures and Algorithms#Computer Science - Discrete Mathematics#Computer Science - Learning#Computer Science - Social and Information Networks#Statistics - Machine Learning

We consider a fundamental algorithmic question in spectral graph theory:
Compute a spectral sparsifier of random-walk matrix-polynomial
$$L_\alpha(G)=D-\sum_{r=1}^d\alpha_rD(D^{-1}A)^r$$ where $A$ is the adjacency
matrix of a weighted, undirected graph, $D$ is the diagonal matrix of weighted
degrees, and $\alpha=(\alpha_1...\alpha_d)$ are nonnegative coefficients with
$\sum_{r=1}^d\alpha_r=1$. Recall that $D^{-1}A$ is the transition matrix of
random walks on the graph. The sparsification of $L_\alpha(G)$ appears to be
algorithmically challenging as the matrix power $(D^{-1}A)^r$ is defined by all
paths of length $r$, whose precise calculation would be prohibitively
expensive.
In this paper, we develop the first nearly linear time algorithm for this
sparsification problem: For any $G$ with $n$ vertices and $m$ edges, $d$
coefficients $\alpha$, and $\epsilon > 0$, our algorithm runs in time
$O(d^2m\log^2n/\epsilon^{2})$ to construct a Laplacian matrix
$\tilde{L}=D-\tilde{A}$ with $O(n\log n/\epsilon^{2})$ non-zeros such that
$\tilde{L}\approx_{\epsilon}L_\alpha(G)$.
Matrix polynomials arise in mathematical analysis of matrix functions as well
as numerical solutions of matrix equations. Our work is particularly motivated
by the algorithmic problems for speeding up the classic Newton's method in
applications such as computing the inverse square-root of the precision matrix
of a Gaussian random field...

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