A new class of linearizations and \$ell\$-ifications for m×m matrix polynomials \$P(x)\$ of degree \$n\$ is proposed. The \$ell\$-ifications in this class have the form \$A(x)=D(x)+(eotimes I_m)W(x)\$ where \$D\$ is a block diagonal matrix polynomial with blocks \$B_i(x)\$ of size \$m\$, \$W\$ is an \$m imes qm\$ matrix polynomial and \$e=(1,…,1)^t in 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^{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.

### On a class of matrix pencils and ℓ-ifications equivalent to a given matrix polynomial

#### Abstract

A new class of linearizations and \$ell\$-ifications for m×m matrix polynomials \$P(x)\$ of degree \$n\$ is proposed. The \$ell\$-ifications in this class have the form \$A(x)=D(x)+(eotimes I_m)W(x)\$ where \$D\$ is a block diagonal matrix polynomial with blocks \$B_i(x)\$ of size \$m\$, \$W\$ is an \$m imes qm\$ matrix polynomial and \$e=(1,…,1)^t in 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^{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.
##### Scheda breve Scheda completa Scheda completa (DC)
2016
Bini, DARIO ANDREA; Robol, Leonardo
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11568/765500`