A new approach to square matrix completion problem

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.