Uniformly one-connected matrices and their inverses

The uniformly one-connected matrices are irreducible matrices A = [aij] such that aii ≠ 0 for i = 1, …, n, and every zero entry of A is contained in a zero s × t submatrix with s + T = n − 1. They include many subclasses of matrices, including among others the unipathic matrices and the upper (lower) Hessenberg matrices. We derive the basic properties of the matrices in this class and show how to construct families of such matrices. We also show that the inverse of uniformly one-connected matrices enjoy many special properties even though they usually have no zero entries. We show how to use these special properties to conclude that a given matrix B is the inverse of a uniformly one-connected matrix.