by Sujit Kumar Mitra* (Indian Statistical Institute, India), P Bhimasankaram (University of Hyderabad, India), & Saroj B Malik (Hindu College, University of Delhi, India)

Table of Contents (396k)
Preface (89k)
Chapter 1: Introduction (185k)

The present monograph on matrix partial orders, the first on this topic, makes 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 Löwner 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 as well as 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.

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• Introduction
• Matrix Decompositions and Generalized Inverses
• The Minus Order
• The Sharp Order
• The Star Order
• One-Sided Orders
• Unified Theory of Matrix Partial Orders through Generalized Inverses
• The Löwner Order
• Parallel Sums
• Schur Complements and Shorted Operators
• Shorted Operators — Other Approaches
• Lattice Properties of Partial Orders
• Partial Orders of Modified Matrices
• Equivalence Relations on Generalized and Outer Inverses
• Applications
• Some Open Problems
• Appendix: Relations and Partial Orders