This book examines two important contemporary areas in linear algebra, namely combinatorial matrix theory and generalized inverses. It covers a wide range of topics of interest such as graph theory, linear algebra, numerical methods and statistical inference.

This book consists of eighteen articles in the area of Combinatorial Matrix Theory' and Generalized Inverses of Matrices'. Original research and expository articles presented in this publication are written by leading Mathematicians and Statisticians working in these areas. The articles contained herein are on the following general topics: matrices in graph theory', generalized inverses of matrices', matrix methods in statistics' and magic squares'. In the area of matrices and graphs, speci_c topics addressed in this volume include energy of graphs, q-analog, immanants of matrices and graph realization of product of adjacency matrices. Topics in the book from `Matrix Methods in Statistics' are, for example, the analysis of BLUE via eigenvalues of covariance matrix, copulas, error orthogonal model, and orthogonal projectors in the linear regression models. Moore-Penrose inverse of perturbed operators, reverse order law in the case of inde_nite inner product space, approximation numbers, condition numbers, idempotent matrices, semiring of nonnegative matrices, regular matrices over incline and partial order of matrices are the topics addressed under the area of theory of generalized inverses. In addition to the above traditional topics and a report on CMTGIM 2012 as an appendix, we have an article on old magic squares from India.

Autorentext

Ravindra B> Bapat is a Professor at the Indian Statistical Institute, Delhi Centre. He is a Fellow of the Indian Academy of Sciences, Indian National Science Academy and a J.C. Bose Fellow. He served as the President of the Indian Mathematical Society during 2007-2008. He has been on the editorial boards of Linear and Multilinear Algebra, Electronic Journal of Linear Algebra, India Journal of Pure and Applied Mathematics and Kerala Mathematical Association Bulletin.

Steve J. Kirkland is a Stokes Professor at the National University of Ireland Maynooth. He is currently an Editor-in-Chief of the Journal Linear and Multilinear Algebra, and serves on the editorial boards of Linear Algebra and its Applications, the Electronic Journal of Linear Algebra, and SIAM Review. He is also the President of the International Linear Algebra Society.

K. Manjunatha Prasad is a Professor of Mathematics at the Department of Statistics, Manipal University, Manipal. He has several publications in the area of his research interest. His research interests are in matrix theory, generalized inverses, ring theory, projective modules and matrices in graph theory.

Simo Puntanen earned his PhD in Statistics from the University of Tampere (Finland) in 1987, where he is presently a Docent and Lecturer of Statistics. He was a Senior Fellow at the Academy of Finland (autumn 1988 and 1993 - 1995), an Associate Professor at Tampere University during 1997 - 1998 and worked as a Senior Scientist of the Academy in 1998 - 1999. He is currently the book review editor of the International Statistical Review.

Inhalt
Skew Spectrum of the Cartesian Product of an Oriented Graph with an Oriented Hypercube.- Notes on Explicit Block Diagonalization.- The Third Immanant of q-Laplacian Matrices of Trees and Laplacians of Regular Graphs.- Matrix Product of Graphs.- Determinant of the Laplacian Matrix of Weighted Directed Graphs .- From Multivariate Skewed Distributions to Copulas.- Revisiting the BLUE in a Linear Model via Proper Eigenvectors.- Inference in Error Orthogonal Models.- On the Entries of Orthogonal Projection Matrices.- Moore-Penrose Inverse of Perturbed Operators on Hilbert Spaces.- The Reverse Order Law in Indefinite Inner Product Spaces.- Generalized Inverses and Approximation Numbers.- On the Level-2 Condition Number for Moore-Penrose Inverse in Hilbert Space.- Products and Sums of Idempotent Matrices over Principal Ideal Domain.- Perfect Semiring of Nonnegative Matrices.- Regular Matrices over an Incline.- Matrix Partial Orders associated with Space Preorder.- An Illustrated Introductionto Some Old Magic Squares from India.- Appendix: Report on CMTGIM 2012, Manipal.- Index