Combinatorial Matrix Theory and Generalized Inverses of Matrices

1. Front Matter

   Pages I-XVII

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2. Skew Spectrum of the Cartesian Product of an Oriented Graph with an Oriented Hypercube

   • A. Anuradha, R. Balakrishnan

   Pages 1-12

3. Notes on Explicit Block Diagonalization

   • Murali K. Srinivasan

   Pages 13-31

4. The Third Immanant of q-Laplacian Matrices of Trees and Laplacians of Regular Graphs

   • R. B. Bapat, Sivaramakrishnan Sivasubramanian

   Pages 33-40

5. Matrix Product of Graphs

   • K. Manjunatha Prasad, G. Sudhakara, H. S. Sujatha, M. Vinay

   Pages 41-55

6. Determinant of the Laplacian Matrix of a Weighted Directed Graph

   • Debajit Kalita

   Pages 57-62

7. From Multivariate Skewed Distributions to Copulas

   • Tõnu Kollo, Anne Selart, Helle Visk

   Pages 63-72

8. Revisiting the BLUE in a Linear Model via Proper Eigenvectors

   • Jan Hauke, Augustyn Markiewicz, Simo Puntanen

   Pages 73-83

9. Inference in Error Orthogonal Models

   • Francisco Carvalho, João Tiago Mexia

   Pages 85-100

10. On the Entries of Orthogonal Projection Matrices

    • Oskar Maria Baksalary, Götz Trenkler

    Pages 101-118

11. Moore–Penrose Inverse of Perturbed Operators on Hilbert Spaces

    • Shani Jose, K. C. Sivakumar

    Pages 119-131

12. The Reverse Order Law in Indefinite Inner Product Spaces

    • Sachindranath Jayaraman

    Pages 133-141

13. Generalized Inverses and Approximation Numbers

    • K. P. Deepesh, S. H. Kulkarni, M. T. Nair

    Pages 143-158

14. On the Level-2 Condition Number for Moore–Penrose Inverse in Hilbert Space

    • Huaian Diao, Yimin Wei

    Pages 159-169

15. Products and Sums of Idempotent Matrices over Principal Ideal Domains

    • K. P. S. Bhaskara Rao

    Pages 171-175

16. Perfect Semiring of Nonnegative Matrices

    • Adel Alahmedi, Yousef Alkhamees, S. K. Jain

    Pages 177-181

17. Regular Matrices over an Incline

    • AR. Meenakshi

    Pages 183-193

18. Matrix Partial Orders Associated with Space Preorder

    • K. Manjunatha Prasad, K. S. Mohana, Y. Santhi Sheela

    Pages 195-226

19. An Illustrated Introduction to Some Old Magic Squares from India

    • George P. H. Styan, Ka Lok Chu

    Pages 227-252

20. A Report on CMTGIM 2012, Manipal

    • R. B. Bapat, K. Manjunatha Prasad

    Pages 253-274

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.

• Energy of Graph
• Generalized Inverse
• Linear Regression Model,
• Magic Square
• Matrix Methods in Graphs
• Partial Order on Matrices
• Spectrum of Graph
• matrix theory

From the reviews:

“This book contains eighteen papers in the field of matrices in graph theory, generalised inverses of matrices, and matrix methods in statistics, written by leading mathematicians and statisticians. … the book covers a wide range of topics with substantial advances and is a welcome volume.” (Shuangzhe Liu, International Statistical Review, Vol. 82 (1), 2014)

• Indian Statistical Institute, New Delhi, India

  Ravindra B. Bapat

• Hamilton Institute, National University of Ireland, Maynooth, Ireland

  Steve J. Kirkland

• Department of Statistics, Manipal University, Manipal, India

  K. Manjunatha Prasad

• , School of Information Sciences, University of Tampere, Tampere, Finland

  Simo Puntanen

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.

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