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.
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. 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.
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. Collecting diverse results into a single volume, this self-contained book highlights the importance and utility of the group inverses of M-matrices in several application areas. The authors emphasize the connections between problems arising in Markov chains, Perron eigenvalue analysis, and spectral graph theory. They show how group inverses offer valuable insight into each of these areas.
Inhaltsverzeichnis
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
Klappentext
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.
