On matrices with Perron--Frobenius properties and some negative entries
The authors study the set \(PF_n\) of \(n \times n\) real matrices for which there is a positive eigenvalue, of multiplicity 1, that strictly dominates all other eigenvalues and for which there is both an entry-wise positive left and right eigenvector. Such matrices may have negative entries and generalize the primitive matrices in important ways. They give a characterization of \(PF_n\) and prove that it is connected. Given a matrix \(A \in PF_n\), the authors consider two circular cones centered at the ray from the origin through \(A\) in matrix space. The first cone, \(C_{\alpha}(A)\), is determined by that matrix angularly nearest to \(A\) that does not have a strictly dominant positive eigenvalue and \(\alpha\) is the cosine of that angle. The second, \(C_{\beta}(A)\), is determined by that matrix angularly nearest to \(A\) that does not have positive eigenvectors associated with any dominant eigenvalue, and \(\beta\) is the corresponding cosine. In the paper, \(\alpha\) and \(\beta\) are calculated in some cases. Finally, the authors describe several ways of constructing this type of matrices, including a very geometric one.
scientific article; zbMATH DE number 2175431