Rank Additivity and Matrix Partial Orders

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


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.


Additional Rank Minus Partial Order Order Stars Parallel Summable Moore-Penrose Inverse 
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Copyright information

© Springer-Verlag London Limited 2012

Authors and Affiliations

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

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