The Rank of the Sum of Two Rectangular Matrices
Canadian mathematical bulletin, Tome 13 (1970) no. 3, p. 384
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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).
Murphy, Ian S. The Rank of the Sum of Two Rectangular Matrices. Canadian mathematical bulletin, Tome 13 (1970) no. 3, p. 384. doi: 10.4153/CMB-1970-072-0
@article{10_4153_CMB_1970_072_0,
author = {Murphy, Ian S.},
title = {The {Rank} of the {Sum} of {Two} {Rectangular} {Matrices}},
journal = {Canadian mathematical bulletin},
pages = {384--384},
year = {1970},
volume = {13},
number = {3},
doi = {10.4153/CMB-1970-072-0},
url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1970-072-0/}
}
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