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- 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

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.

**Motivation and Examples **An example from population modelling

An example from Markov chains

**The Group Inverse **Definition and general properties of the group inverse

Spectral properties of the group inverse

Expressions for the group inverse

Group inverse versus Moore–Penrose inverse

The group inverse associated with an M-matrix

**Group Inverses and Derivatives of Matrix Functions **Eigenvalues as functions

First and second derivatives of the Perron value

Concavity and convexity of the Perron value

First and second derivatives of the Perron vector

**Perron Eigenpair in Demographic Applications **Introduction to the size-classified population model

First derivatives for the stage-classified model

Second derivatives of the Perron value in the age-classified model

Elasticity and its derivatives for the Perron value

**The Group Inverse in Markov Chains **Introduction to Markov chains

Group inverse in the periodic case

Perturbation and conditioning of the stationary distribution vector

Bounds on the subdominant eigenvalue

Examples

**Mean First Passage Times for Markov Chains**

Mean first passage matrix via the group inverse

A proximity inequality for the group inverse

The inverse mean first passage matrix problem

A partitioned approach to the mean first passage matrix

The Kemeny constant

**Applications of the Group Inverse to Laplacian Matrices **Introduction to the Laplacian matrix

Distances in weighted trees

Bounds on algebraic connectivity via the group inverse

Resistance distance, the Weiner index and the Kirchhoff index

Interpretations for electrical networks

**Computing the Group Inverse **Introduction

The shuffle and Hartwig algorithms

A divide and conquer method

Stability issues for the group inverse

**Bibliography**

**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.