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## Relative asymptotics for orthogonal matrix polynomials

Branquinho, A.; Marcellán, F.; Mendes, A.
Tipo: Pré-impressão
ENG
Relevância na Pesquisa
56.42%
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.

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

Yakhlef, Hossain O.; Marcellán, Francisco
Tipo: Artigo de Revista Científica Formato: application/pdf
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

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

Yakhlef, Hossain O.; Marcellán, Francisco; Piñar, Miguel A.
Tipo: Artigo de Revista Científica
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

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

Pereira, E.; Vitória, J.
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

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

Bazán,Fermin S. Viloche
Tipo: Artigo de Revista Científica Formato: text/html
Relevância na Pesquisa
56.25%
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.

## Matrix polynomials with completely prescribed eigenstructure

Terán Vergara, Fernando de; Martínez Dopico, Froilan C.; Van Dooren, Paul
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
Relevância na Pesquisa
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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).

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

Terán Vergara, Fernando de; Martínez Dopico, Froilan C.; Mackey, Don Steven
Tipo: info:eu-repo/semantics/acceptedVersion; info:eu-repo/semantics/article
Relevância na Pesquisa
46.22%
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.

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

Bueno, M. I.; Martínez Dopico, Froilan C.; Furtado, S.; Rychnovsky, M.
Tipo: info:eu-repo/semantics/acceptedVersion; info:eu-repo/semantics/article
Relevância na Pesquisa
56.22%
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...

## Relative asymptotics for orthogonal matrix polynomials

Branquinho, A.; Marcellán, F.; Mendes, A.
Tipo: Artigo de Revista Científica
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.

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

Borrego, Jorge; Castro, Mirta; Durán, Antonio J.
Tipo: Artigo de Revista Científica
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

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

Klep, Igor; Schweighofer, Markus
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.3%
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

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

Kokabifar, E.; Loghmani, G. B.; Psarrakos, P. J.
Tipo: Artigo de Revista Científica
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.

## Orthogonal matrix polynomials and higher order recurrence relations

Durán, Antonio J.; Van Assche, Walter
Tipo: Artigo de Revista Científica
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.

## Backward errors and linearizations for palindromic matrix polynomials

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

## On Pseudospectra of Matrix Polynomials and their Boundaries

Boulton, Lyonell; Lancaster, Peter; Psarrakos, Panayiotis
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

## Matrix polynomials, generalized Jacobians, and graphical zonotopes

Izosimov, Anton
Tipo: Artigo de Revista Científica
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

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

Kokabifar, Esmaeil; Loghmani, G. B.; Nazari, A. M.; Karbassi, S. M.
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.

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

Duran, Antonio J.; de la Iglesia, Manuel D.
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

## Locating the eigenvalues of matrix polynomials

Bini, Dario A.; Noferini, Vanni; Sharify, Meisam
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

## Spectral Sparsification of Random-Walk Matrix Polynomials

Cheng, Dehua; Cheng, Yu; Liu, Yan; Peng, Richard; Teng, Shang-Hua
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...