Uniformly one-connected matrices and their inverses (Q1375096)

An \(n\times n\) irreducible matrix is uniformly one-connected if the diagonal elements are non-zero and every zero entry is contained in a zero \(s\times t\) submatrix with \(s+t= n-1\). Examples are the unipathic matrices and the upper (lower) Hessenberg matrices. The basic properties of uniformly one-connected matrices are derived and it is shown how to construct families of such matrices. Special properties of the inverses of uniformly one-connected matrices are used to test whether a given matrix \(B\) is the inverse of a uniformly one-connected matrix.