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.
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© 2012 Springer-Verlag London Limited
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Bapat, R.B. (2012). Rank Additivity and Matrix Partial Orders. In: Linear Algebra and Linear Models. Universitext. Springer, London. https://doi.org/10.1007/978-1-4471-2739-0_6
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DOI: https://doi.org/10.1007/978-1-4471-2739-0_6
Publisher Name: Springer, London
Print ISBN: 978-1-4471-2738-3
Online ISBN: 978-1-4471-2739-0
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