We develop a Hurwitz stability criterion for nonmonic matrix polynomials via column reduction, generalizing existing approaches constrained by the monic assumption, as well as Gantmacher's classical stability criterion via Markov parameters. Starting from redefining the associated Markov parameters through a column-wise adaptive splitting method, our framework constructs two structured matrices whose rectangular Hankel blocks are obtained via the extraction of these parameters. We establish an explicit interrelation between the inertias of column reduced matrix polynomials and the derived structured matrices. Furthermore, we demonstrate that the Hurwitz stability of column reduced matrix polynomials can be determined by the Hermitian positive definiteness of these rectangular block Hankel matrices.

翻译：本文通过列约化方法建立了非首一矩阵多项式的Hurwitz稳定性判据，推广了现有受限于首一假设的方法以及Gantmacher基于马尔可夫参数的经典稳定性判据。通过列自适应分割方法重新定义相关马尔可夫参数，本框架构建了两个结构化矩阵，其矩形Hankel分块通过提取这些参数获得。我们建立了列约化矩阵多项式的惯性特征与导出结构化矩阵之间的显式关联。进一步证明，列约化矩阵多项式的Hurwitz稳定性可由这些矩形分块Hankel矩阵的Hermitian正定性判定。