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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

67.04%

#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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## On a Class of Matrix Pencils and $\ell$-ifications Equivalent to a Given Matrix Polynomial

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

16.08%

A new class of linearizations and $\ell$-ifications for $m\times m$ matrix
polynomials $P(x)$ of degree $n$ is proposed. The $\ell$-ifications in this
class have the form $A(x) = D(x) + (e\otimes I_m) W(x)$ where $D$ is a block
diagonal matrix polynomial with blocks $B_i(x)$ of size $m$, $W$ is an $m\times
qm$ matrix polynomial and $e=(1,\ldots,1)^t\in\mathbb C^q$, for a suitable
integer $q$. The blocks $B_i(x)$ can be chosen a priori, subjected to some
restrictions. Under additional assumptions on the blocks $B_i(x)$ the matrix
polynomial $A(x)$ is a strong $\ell$-ification, i.e., the reversed polynomial
of $A(x)$ defined by $A^\#(x) := x^{\mathrm{deg} A(x)} A(x^{-1})$ is an
$\ell$-ification of $P^\#(x)$. The eigenvectors of the matrix polynomials
$P(x)$ and $A(x)$ are related by means of explicit formulas. Some practical
examples of $\ell$-ifications are provided. A strategy for choosing $B_i(x)$ in
such a way that $A(x)$ is a well conditioned linearization of $P(x)$ is
proposed. Some numerical experiments that validate the theoretical results are
reported; Comment: 20 pages, 8 figures

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