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Rank Additivity and Matrix Partial Orders

  • R. B. Bapat
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.

Copyright information

© Springer-Verlag London Limited 2012

Authors and Affiliations

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

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