# Group Inverses of M-Matrices and Their Applications

Stephen J. Kirkland, Michael Neumann

December 18, 2012 by Chapman and Hall/CRC
Reference - 332 Pages - 23 B/W Illustrations
ISBN 9781439888582 - CAT# K14074
Series: Chapman & Hall/CRC Applied Mathematics & Nonlinear Science

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### Features

• Brings together a diverse collection of results from the past 30 years on the study and application of nonnegative matrices and M-matrices
• Applies a variety of techniques, including tools from analysis, graph theory, and numerical linear algebra
• Explores the connections between problems arising in Markov chains, Perron eigenvalue analysis, and spectral graph theory
• Explains how the group inverse can be computed using shuffle and Hartwig algorithms and a divide and conquer method
• Assumes some familiarity with nonnegative matrices, directed and undirected graphs, and Markov chains

### Summary

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.