Group Inverses of M-Matrices and Their Applications

1. Auflage 2012
ISBN: 978-1-4398-8858-2
Verlag: Taylor & Francis Inc

Buch, Englisch, 332 Seiten, Format (B × H): 159 mm x 241 mm, Gewicht: 612 g

Reihe: Chapman & Hall/CRC Applied Mathematics & Nonlinear Science

Group inverses for singular M-matrices are useful tools not only in matrix analysis, but also in the analysis of stochastic processes, graph theory, electrical networks, and demographic models. Group Inverses of M-Matrices and Their Applications highlights the importance and utility of the group inverses of M-matrices in several application areas.

After introducing sample problems associated with Leslie matrices and stochastic matrices, the authors develop the basic algebraic and spectral properties of the group inverse of a general matrix. They then derive formulas for derivatives of matrix functions and apply the formulas to matrices arising in a demographic setting, including the class of Leslie matrices. With a focus on Markov chains, the text shows how the group inverse of an appropriate M-matrix is used in the perturbation analysis of the stationary distribution vector as well as in the derivation of a bound for the asymptotic convergence rate of the underlying Markov chain. It also illustrates how to use the group inverse to compute and analyze the mean first passage matrix for a Markov chain. The final chapters focus on the Laplacian matrix for an undirected graph and compare approaches for computing the group inverse.

Collecting diverse results into a single volume, this self-contained book emphasizes the connections between problems arising in Markov chains, Perron eigenvalue analysis, and spectral graph theory. It shows how group inverses offer valuable insight into each of these areas.

Zielgruppe

Professional Practice & Development

Autoren/Hrsg.

Kirkland, Stephen J.

Neumann, Michael

Fachgebiete

• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Angewandte Mathematik, Mathematische Modelle
• Mathematik | Informatik Mathematik Algebra Lineare und multilineare Algebra, Matrizentheorie
• Naturwissenschaften Physik Physik Allgemein Theoretische Physik, Mathematische Physik, Computerphysik
• Technische Wissenschaften Technik Allgemein Mathematik für Ingenieure

Inhaltsverzeichnis

Motivation and Examples. The Group Inverse. Group Inverses and Derivatives of Matrix Functions. Perron Eigenpair in Demographic Applications. The Group Inverse in Markov Chains. Mean First Passage Times for Markov Chains. Applications of the Group Inverse to Laplacian Matrices. Computing the Group Inverse. Bibliography.

Über Autor(innen)

Stephen J. Kirkland is a Stokes Professor at the National University of Ireland, Maynooth. He is editor-in-chief of Linear and Multilinear Algebra and serves on the editorial boards of several other journals. Dr. Kirkland’s research interests are primarily in matrix theory and graph theory, with an emphasis on the interconnections between these two areas.

Michael Neumann was the Stuart and Joan Sidney Professor of Mathematics and a Board of Trustees Distinguished Professor at the University of Connecticut. Dr. Neumann published more than 160 mathematical papers, mainly in matrix theory, numerical linear algebra, and numerical analysis.