Patterns of commutativity: the commutant of the full pattern,

Identified are a number of conditions on square patterns that are closely related to allowing commutativity with the full pattern. Implications and examples that show non-implications are given, along with a graph that summarizes the provided information. A complete description of commutativity with the full pattern is given in both the irreducible case and the reducible case in which there are two irreducible components. 1. Introduction, problem statement and notation. By a pattern P (re- spectively, sign pattern S )w e mean an array of∗'s and 0's (+'s, −'s and 0's) in which a ∗ (+ or −) indicates a nonzero (positive or negative) entry. A real matrix A =( ai,j) belongs to pattern P (sign pattern S) if its dimensions agree with those of P (S )a nd ai,jif and only if the i, j entry of P is a ∗ (ai,j > 0, ai,j < 0i f ft hei, j entry of S is, respectively, + or −). We say that two n-by-n patterns P and Q (a pattern P and a sign pattern S) commute (or allow commutativity) if there exist matrices A ∈P , B ∈Q ( ∈S ) that commute, i.e., AB = BA. In general we say that a pattern allows a given property if there exists a matrix of the pattern with that property (e. g. we are considering pairs of patterns that allow commutativity); a pattern requires a given property if every matrix of the pattern has that property. The commutant of a pattern P (sign pattern S) is simply the set of all patterns Q that commute with P (S). Let C (P )( C (S)) denote the commutant of P (S). Our interest here lies in determining the commutant of the full (all ∗'s) pattern F and of the all + sign pattern J. Of course C (J ) ⊆ C (F ), but, as we will see later, the opposite inclusion is not true. We begin with a discussion of (new) conditions that are necessary for a pattern to be in C (F ), then identify several conditions (some familiar) that are sufficient and identify implications (and non-implications) among these. We also discuss necessary and sufficient conditions in terms of the number of components in the Frobenius normal form of P. Many matrix concepts and notation, such as irreducibility, submatrices, and mul- tiplication by a permutation or diagonal matrix, extend in an unambiguous way to patterns, and we use them in the context of patterns without comment.