Sign patterns that require exactly one real eigenvalue and patterns that requiren−1 nonreal eigenvalues
Carolyn A. Eschenbach · Linear and Multilinear Algebra · 1993
We characterize the class π of all odd dimensional n-by-n sign pattern matrices that require exactly one real eigenvalue (exactly n−1 nonreal eigenvalues). Since more restrictions are needed on the odd cycles in the sign singular patterns in π, and in patterns that allow singularity in π, we give our characterization on three theorems; namely, one theorem to characterize the sign singular patterns in π; one theorem to characterize the patterns in π that allow singularity; and, finally, one theorem to characterize the sign nonsingular patterns in π. Next we combine our results with a previously established characterization of the sign patterns that require n nonreal eigenvalues, and we characterize the sign patterns that require, at leastn−1 nonreal eivenvalues.