Matrix Partial Orders Shorted Operators And Applications
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Matrix Partial Orders, Shorted Operators and Applications
The present monograph on matrix partial orders, appearing for the first time, is a unique presentation of many partial orders on matrices that have fascinated mathematicians for their beauty and applied scientists for their wide-ranging application potential. Except for the Lwner order, the partial orders considered are relatively new and came into being in the late 1970s. After a detailed introduction to generalized inverses and decompositions, the three basic partial orders namely, the minus, the sharp and the star and the corresponding one-sided orders are presented using various generalized inverses. The authors then give a unified theory of all these partial orders. This is followed by a study of the Lwner order and a limited treatment of majorization (there is an abundance of literature available on majorization). The authors also study the parallel sums and shorted matrices, the latter being studied at great length. Partial orders of modified matrices are a new addition. Finally, applications are given in statistics and electrical network theory.
Matrix Partial Orders, Shorted Operators and Applications
1. Introduction. 1.1. Matrix orders. 1.2. Parallel sum and shorted operator. 1.3. A tour through the rest of the monograph -- 2. Matrix decompositions and generalized inverses. 2.1. Introduction. 2.2. Matrix decompositions. 2.3. Generalized inverse of a matrix. 2.4. The group inverse. 2.5. Moore-Penrose inverse. 2.6. Generalized inverses of modified matrices. 2.7. Simultaneous diagonalization. 2.8. Exercises -- 3. The minus order. 3.1. Introduction. 3.2. Space pre-order. 3.3. Minus order - some characterizations. 3.4. Matrices above/below a given matrix under the minus order. 3.5. Subclass of g-inverses A[symbol] of A such that [symbol]A = A[symbol]B and AA[symbol]=BA[symbol] when A
Combinatorial Matrix Theory and Generalized Inverses of Matrices
Author: Ravindra B. Bapat
language: en
Publisher: Springer Science & Business Media
Release Date: 2013-02-11
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.