Rank conditions for sign patterns that allow diagonalizability

Abstract

It is known that for each k≥4, there exists an irreducible sign pattern with minimum rank k that does not allow diagonalizability. However, it is shown in this paper that every square sign pattern A with minimum rank 2 that has no zero line allows diagonalizability with rank 2 and also with rank equal to the maximum rank of the sign pattern. In particular, every irreducible sign pattern with minimum rank 2 allows diagonalizability. On the other hand, an example is given to show the existence of a square sign pattern with minimum rank 3 and no zero line that does not allow diagonalizability; however, the case for irreducible sign patterns with minimum rank 3 remains open. In addition, for a sign pattern that allows diagonalizability, the possible ranks of the diagonalizable real matrices with the specified sign pattern are shown to be lengths of certain composite cycles. Some results on sign patterns with minimum rank 2 are extended to sign pattern matrices whose maximal zero submatrices are “strongly disjoint” (that is, their row index sets as well as their column index sets are pairwise disjoint).