Constructions for potentially stable sign patterns

A square sign pattern A is potentially stable (PS) if there exists a real matrix having the sign pattern A and with all its eigenvalues having negative real parts. The characterization of PS sign patterns remains a long standing open problem. Here three techniques are given for the construction of larger order PS sign patterns from given PS sign patterns. These techniques are: construction of a sign pattern with a nested sequence of properly signed principal minors (a nest), bordering of a PS sign pattern, and use of a similarity transformation. The minimum number of nonzero entries in an irreducible PS sign pattern is determined for small orders and for an arbitrary sign pattern that allows a nest. For sign patterns of order at least four, a bordering construction leads to a new upper bound for this minimum number.