In this paper, we plan to show an eigenvalue algorithm for block Hessenberg matrices by using the idea of non-commutative integrable systems and matrix-valued orthogonal polynomials. We introduce adjacent families of matrix-valued $\theta$-deformed bi-orthogonal polynomials, and derive corresponding discrete non-commutative hungry Toda lattice from discrete spectral transformations for polynomials. It is shown that this discrete system can be used as a pre-precessing algorithm for block Hessenberg matrices. Besides, some convergence analysis and numerical examples of this algorithm are presented.

翻译：本文旨在利用非交换可积系统与矩阵值正交多项式的思想，给出块Hessenberg矩阵的特征值算法。我们引入了矩阵值$\theta$-变形双正交多项式的相邻族，并从多项式的离散谱变换推导出相应的离散非交换饥饿Toda格。研究表明，该离散系统可作为块Hessenberg矩阵的预处理算法。此外，本文还给出了该算法的收敛性分析及数值算例。