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The Rank of the Sum of Two Rectangular Matrices

Published online by Cambridge University Press:  20 November 2018

Ian S. Murphy
Ian S. Murphy*
Affiliation:
University of Edinburgh, Edinburgh, Scotland
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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).

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 13 , Issue 3 , September 1970 , pp. 384
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Meyer, C. D., On the rank of the sum of two rectangular matrices, Canad. Math. Bull. 12 (1969), 508.Google Scholar
2. Mirsky, L., An Introduction to linear algebra, Oxford, 1955.Google Scholar