On general matrices having the Perron-Frobenius Property

Abstract

A matrix is said to have the Perron-Frobenius property if its spectral radius is an
eigenvalue with a corresponding nonnegative eigenvector. Matrices having this and similar properties are studied in this paper as generalizations of nonnegative matrices. Sets consisting of such generalized nonnegative matrices are studied and certain topological aspects such as connectedness and closure are proved. Similarity transformations leaving such sets invariant are completely described, and it is shown that a nonnilpotent matrix eventually capturing the Perron-Frobenius propertyis in fact a matrix that already has it.