Frank J. Hall

On the period and base of a sign pattern matrix

Abstract A square sign pattern matrix A (whose entries are +, −, or 0) is said to be powerful if all the powers A 1 , A 2 , A 3 ,…, are unambiguously defined. For a powerful pattern A , if A l = A l + p with l and p minimal, then l is called the base of A and p is called the period of A . We characterize irreducible powerful sign pattern matrices and investigate the period and base of a powerful sign pattern matrix. We also consider some connections with real matrices and give some significant classes of powerful patterns.

The matrix equation A X – XB = C and its special cases

The consistency and solutions of the matrix equations

An Interlacing Result on Normalized Laplacians

A\bar X - XB = C

Symmetric sign pattern matrices that require unique inertia

,

Generalized Inverses of a Bordered Matrix of Operators

A\bar X \pm XA^T = C

Reducible powerful ray pattern matrices

,

On Ranks of Matrices Associated with Trees

A\bar X \pm XA^* = C

Sign patterns with minimum rank 2 and upper bounds on minimum ranks

are characterized. As a consequence it is shown that

Sign patterns that allow eventual positivity

A^T

Further Results on Generalized Inverses of Partitioned Matrices

(respectively,