The Rank of the Sum of Two Rectangular Matrices
Canadian mathematical bulletin, Tome 13 (1970) no. 3, p. 384
Voir la notice de l'article provenant de la source Cambridge University Press
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/}
}
[1] 1. Meyer, C. D., On the rank of the sum of two rectangular matrices, Canad. Math. Bull. 12 (1969), 508. Google Scholar
[2] 2. Mirsky, L., An Introduction to linear algebra, Oxford, 1955. Google Scholar
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