On the Spectra of Striped Sign Patterns
Judith J. McDonald, Dale D. Olesky, Michael J. Tsatsomeros, Pauline Van den Driessche · Linear and Multilinear Algebra · 2003
Sign patterns consisting of some positive and some negative columns, with at least one of each kind, are shown to allow any self-conjugate spectrum, and thus to allow any inertia. In the case of the n × n sign pattern with all columns positive, given any self-conjugate multiset consisting of n m 1 complex numbers supplemented by a sufficiently large positive number, it is shown how to construct a positive normal matrix whose spectrum is this multiset. Thus, the positive sign pattern allows any inertia with at least one positive eigenvalue.