On Nonnegative Matrices and Generalized M-matrices

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This dissertation deals mainly with analyzing and characterizing certain classes of matrices, non-negative matrices and generalizations of M-matrices. More speci cally, this involves a detailed study of the structure of the generalized eigenspace of non-negative matrices and the generalized nullspace of generalized M-matrices. We rst use preferred and quasi-preferred bases of generalized eigenspaces associated with the spectral radius of non-negative matrices to analyze the existence and uniqueness of a variant of the Jordan canonical form which we call a Frobenius-Jordan form. Based on the Frobenius-Jordan form, spectral and combinatorial properties of nonnegative matrices are discussed. We also consider graph representations of nonnegative bases for nonnegative matrices and derived necessary conditions for the existence of such graph bases. Next we consider two types of generalized M-matrices based on the generalization of nonnegative matrices, namely the class of GM-matrices and M_- matrices. In this thesis we attempt to generalize the combinatorial properties of singular M-matrices to the class of singular GM-matrices and singular M_-matrices. We prove the existence of a preferred basis for a subclass of M_-matrices and obtain similar equivalent conditions for the equality of the height and level characteristics of M_-matrices. In an attempt to obtain similar results for the class of GM-matrices, we give a complete answer regarding the existence of a preferred basis in terms of the order of such matrices. We also provide some characterizations of nonsingular M_-matrix involving positivity of the sums of principal minors and stability. We show that some of the important properties, such as inverse positivity, do not carry over to the entire class of M_-matrices, but to a subclass of these matrices. We also extend the inverse-positivity property of nonsingular M_-matrices to generalized inverses of singular M_-matrices. Motivated by interesting characterizations of singular M-matrices, we then introduce the concepts of eventually monotonicity and eventually nonnegativity on subsets of Rn, which are used to characterize a subclass of matrices.
Supervisor: Sriparna Bandopadhyay