Self-Inverse Sign Patterns

We first show that if A is a self-inverse sign pattern matrix, then every principal submatrix of A that is not combinatorially singular (does not require singularity) is also a self-inverse sign pattern. Next we characterize the class of all n-by-n irreducible self-inverse sign pattern matrices. We then discuss reducible self-inverse patterns, assumed to be in Frobenius normal form, in which each irreducible diagonal block is a self-inverse sign pattern matrix. Finally we present an implicit form for determining the sign patterns of the off-diagonal blocks (unspecified block matrices) so that a reducible matrix is self-inverse.