Flows on graphs applied to diagonal similarity and diagonal equivalence for matrices

Three equivalence relations are considered on the set of n x n matrices with elements in Fo' an · abelian group with absorbing zero adjoined. They are the relations of diagonal similarity, diagonal equivalence, and restricted diagonal equivalence. These relations are usually considered for matrices with elements in a field, but only multiplication is involved. Thus our formulation in terms of an abelian group with 0 is natural. Moreover, if F is chosen to be an additive group, diagonal similarity is characterized in terms of flows on the pattern graph of the matrices and diagonal equivalence in terms of flows on the bipartite graph of the matrices. For restricted diagonal equivalence a pseudo-diagonal of the graph must also be considered. When no pseudo-diagonal is present, the divisibility properties of the grO\iP F playa role. We show that the three relations are characterized by cyclic, polygonal, and pseudo-diagonal products for multiplicative F. Thus, our method of reducing propositions concering the three equivalence relations to propositions concerning flows on graphs, provides a unified approach to problems previously considered independently, and yields some new or improved results. Our consideration of cycles rather than circuits eliminates certain restrictions (e.g., the complete reducibility of the matrices) which have previously been imposed. Our results extend theorems in Engel and Schneider [5], where however the group F is permitted to be non-commutative.