Linear Algebra and Linear Models pp 51-60 | Cite as

# Rank Additivity and Matrix Partial Orders

## 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.