In the framework of Polynomial Eigenvalue Problems, most of the matrix polynomials arising in applications are structured polynomials (namely (skew-)symmetric, (skew-)Hermitian, (anti-)palindromic, or alternating). The standard way to solve Polynomial Eigenvalue Problems is by means of linearizations. The most frequently used linearizations belong to general constructions, valid for all matrix polynomials of a fixed degree, known as {\em companion linearizations}. It is well known, however, that is not possible to construct companion linearizations that preserve any of the previous structures for matrix polynomials of even degree. This motivates the search for more general companion forms, in particular {\em companion $\ell$-ifications}. In this paper, we present, for the first time, a family of (generalized) companion $\ell$-ifications that preserve any of these structures, for matrix polynomials of degree $k=(2d+1)\ell$. We also show how to construct sparse $\ell$-ifications within this family. Finally, we prove that there are no structured companion quadratifications for quartic matrix polynomials.

翻译：在多面性电子价值问题的框架内,应用中产生的多数矩阵多元数字是结构化的多面性(即(skew-)对称,(skew-)Hermitian,(ant-)palindromic,或交替)。解决多面性电子价值问题的标准方法是线性化。最常用的线性化属于一般结构,适用于固定程度的所有矩阵多面性多面性,称为 $(mem) 相伴线性。然而,众所周知,不可能建立保护任何先前的多面性矩阵结构的相伴线性。这促使人们寻找更一般的同伴形式,特别是$(ell-美元)加分义化。本文首次介绍一个(通用的)配方$($\ell-美元)的组合,用于保存这些结构的任何结构。对于 美元=(2d+1)\ell\ell等组合中的任何组合,我们最后也展示了如何构建这个结构化的组合。