| General matrices with a positive dominant eigenvalue and a corresponding nonnegative eigenvector are studied. Such matrices are said to possess the Perron-Frobenius property. The latter properly is naturally enjoyed by nonnegative matrices and has a wide variety of applications. In this dissertation, general matrices, which are not necessarily nonnegative, that possess the Perron-Frobenius property are analyzed. Several characterizations of matrices having the Perron-Frobenius property are presented: spectral, combinatorial, and geometric characterizations. In some cases, a full characterization is obtained, while in others only certain aspects are studied. In addition, some combinatorial, topological and spectral properties of matrices enjoying the Perron-Frobenius property are presented and the similarity transformations preserving the Perron-Frobenius property are completely described. Furthermore, generalizations of M-matrices are studied, including the new class of GM-matrices. Matrices in the latter class are of the form sI--B where B and its transpose possess the Perron-Frobenius property and the spectral radius of B is less than s. Results analogous to those known for M-matrices are demonstarted. Also, various splittings of GM-matrices are studied along with conditions for their convergence. |
