Group Inverses of M-Matrices and Their Applications

Group Inverses of M-Matrices and Their Applications

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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.

Table of Contents

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

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
The shuffle and Hartwig algorithms
A divide and conquer method
Stability issues for the group inverse


Author Bio(s)

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.

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