ABSTRACT

The classical completion problem of describing the possible similarity class of a square matrix with a prescribed arbitrary submatrix was studied by many authors through time, and it is completely solved in [Dodig M, Stošić M. Similarity class of a matrix with prescribed submatrix. Linear Multilinear Algebra. 2009;57:217–245; Combinatorics of polynomial chains. Linear Algebra Appl. 2020;589:130–157]. In this paper we show a surprising relation between this notable problem and the problem of describing the feedback invariants of restrictions and quotients of series connected systems studied in [Baragaña I, Zaballa I. Feedback invariants of restrictions and quotients: series connected systems. Linear Algebra Appl. 2002;351-352:69–89; Dodig M, Silva FC. Controllability of series connections of arbitrarily many linear systems. Linear Algebra Appl. 2008;429:122–141]. As a corollary, we obtain a new combinatorial result on partitions of integers.

摘要

许多作者通过时间研究了描述具有指定任意子矩阵的方阵的可能相似性类的经典完成问题，并在 [Dodig M, Stošić M. 具有指定子矩阵的矩阵的相似性类中完全解决。线性多线性代数。2009；57：217–245；多项式链的组合。线性代数应用程序。2020；589:130–157]。在本文中，我们展示了这个值得注意的问题与描述串联连接系统的限制和商的反馈不变量的问题之间的惊人关系，该问题在 [Baragaña I, Zaballa I. 限制和商的反馈不变量：串联系统中研究。线性代数应用程序。2002；351-352：69–89；多迪格 M，席尔瓦足球俱乐部。任意多线性系统串联的可控性。线性代数应用程序。2008；429:122–141]。作为推论，我们获得了一个关于整数分区的新组合结果。