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When Does Rank(A+B)=Rank(A)+Rank(B)?

Published online by Cambridge University Press:  20 November 2018

G. Marsaglia  and
G. P. H. Styan
G. Marsaglia
Affiliation:
McGill University, Montreal, Quebec
G. P. H. Styan
Affiliation:
McGill University, Montreal, Quebec
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In a recent note in the Bulletin, Murphy [5] gave a short proof that for complex m×n matrices A and B, r(A+B)=r(A)+r(B) if the rows of A are orthogonal to the rows of B and the columns of A are orthogonal to the columns of B. His proof was elegant and simple, an improvement on an earlier proof of the same result by Meyer [4].

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 15 , Issue 3 , September 1972 , pp. 451 - 452
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Khatri, C. G., A simplified approach to the derivation of the theorems on the rank of a matrix, J. Maharaja Sayajirao Univ. Baroda, 10 (1961), 1-5.Google Scholar
2. George, Marsaglia, Bounds on the rank of the sum of matrices, Trans, of the Fourth Prague Conf. on Information Theory, Statistical Decision Functions, Random Processes (Prague, August 31-Sept. 11, 1965), Czechoslovak Acad. Sci. (1967), 455-462.Google Scholar
3. George, Marsaglia and Styan, George P. H., Inequalities and equalities for ranks of matrices (to appear).Google Scholar
4. Meyer, C. D., On the rank of the sum of two rectangular matrices, Canad. Math. Bull. 12 (1969), p. 508.Google Scholar
5. Ian S., Murphy, The rank of the sum of two rectangular matrices, Canad. Math. Bull. 13 (1970), p. 384.Google Scholar