On a class of matrix pencils and ℿ/span>-ifications equivalent to a given matrix polynomial

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• 作者：Dario A. Bini^a ; ^{bini@dm.unipi.it" class="auth_mail" title="E-mail the corresponding author} ; Leonardo Robol^b ; ^{leonardo.robol@sns.it" class="auth_mail" title="E-mail the corresponding author}
• 关键词：65F15 ; 15A21 ; 15A03
• 刊名：Linear Algebra and its Applications
• 出版年：2016
• 出版时间：1 August 2016
• 年：2016
• 卷：502
• 期：Complete
• 页码：275-298
• 全文大小：701 K

文摘

A new class of linearizations and ℓ  -ifications for m×m$m\times m$ matrix polynomials a66970a0e88c4cdd" title="Click to view the MathML source">P(x)$P(x)$ of degree n is proposed. The ℓ  -ifications in this class have the form A(x)=D(x)+(e⊗I_m)W(x)$A(x)=D(x)+(e\otimes I_m)W(x)$ where D   is a block diagonal matrix polynomial with blocks B_i(x)$B_i(x)$ of size m, W   is an m×qm$m\times qm$ matrix polynomial and e=(1,…,1)^t∈C^q$e={(1,\dots ,1)}^t\in {\mathbb{C}}^q$, for a suitable integer q  . The blocks B_i(x)$B_i(x)$ can be chosen a priori, subjected to some restrictions. Under additional assumptions on the blocks B_i(x)$B_i(x)$ the matrix polynomial A(x)$A(x)$ is a strong ℓ  -ification, i.e., the reversed polynomial of A(x)$A(x)$ defined by A^#(x):=x^deg⁡A(x)A(x⁻¹)$A^{\mathrm{\#}}(x):=x^{\mathrm{deg}A(x)}A(x^{-1})$ is an ℓ  -ification of P^#(x)$P^{\mathrm{\#}}(x)$. The eigenvectors of the matrix polynomials a66970a0e88c4cdd" title="Click to view the MathML source">P(x)$P(x)$ and A(x)$A(x)$ are related by means of explicit formulas. Some practical examples of ℓ  -ifications are provided. A strategy for choosing B_i(x)$B_i(x)$ in such a way that A(x)$A(x)$ is a well conditioned linearization of a66970a0e88c4cdd" title="Click to view the MathML source">P(x)$P(x)$ is proposed. Some numerical experiments that validate the theoretical results are reported.