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Rank Additivity and Matrix Partial Orders

  • R. B. Bapat
Chapter
Part of the Universitext book series (UTX)

Abstract

If A, B are m×n matrices then \(\operatorname{rank} B \le \operatorname{rank} A + \operatorname{rank}(B-A)\). When does equality hold in this inequality? This question is related to many notions such as g-inverses, minus partial order and parallel sum. The phenomenon is known as rank additivity. We first prove a characterization result which brings together several conditions equivalent to rank \(B = \operatorname{rank} A + \operatorname{rank} (B-A)\). We then introduce the star order, a partial order on matrices which is a refinement of the minus order. Basic properties of the star order bringing out its relation with the Moore–Penrose inverse are proved.

Keywords

Additional Rank Minus Partial Order Order Stars Parallel Summable Moore-Penrose Inverse 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Copyright information

© Springer-Verlag London Limited 2012

Authors and Affiliations

  • R. B. Bapat
    • 1
  1. 1.Indian Statistical InstituteNew DelhiIndia

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