On Matrices with Perron–Frobenius Properties and Some Negative Entries

We consider those n-by-n matrices with a strictly dominant positive eigenvalue of multiplicity 1 and associated positive left and right eigenvectors. Such matrices may have negative entries and generalize the primitive matrices in important ways. Several ways of constructing such matrices, including a very geometric one, are discussed. This paper grew out of a recent survey talk about nonnegative matrices by the first author and a joint paper, with others, by the second author about the symmetric case [Tarazaga et al. (2001) Linear Algebra Appl. 328: 57].