The Rank of the Sum of Two Rectangular Matrices

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Abstract
In what follows, the transposed complex conjugate of a complex rectangular matrix D is denoted by D* and the rank of D by r(D). Meyer [1] proved the following result using generalized inverses:Theorem. Let A and B be complex m × n matrices such that AB*=B*A=0. Then r(A+B) = r(A)+r(B).Below we prove this result by repeated use of the fact that for every complex m × n matrix D we have r(D) = r(D*D) = r(DD*) (e.g. See [2] Theorem 5.5.4).

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