Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-26T13:32:53.146Z Has data issue: false hasContentIssue false
  • English
  • Français

On generalized inverses of matrices

Published online by Cambridge University Press:  24 October 2008

M. H. Pearl
M. H. Pearl
Affiliation:
Imperial College, London and the University of Maryland, College Park
Get access Rights & Permissions [Opens in a new window]

Extract

The notion of the inverse of a matrix with entries from the real or complex fields was generalized by Moore (6, 7) in 1920 to include all rectangular (finite dimensional) matrices. In 1951, Bjerhammar (2, 3) rediscovered the generalized inverse for rectangular matrices of maximal rank. In 1955, Penrose (8, 9) independently rediscovered the generalized inverse for arbitrary real or complex rectangular matrices. Recently, Arghiriade (1) has given a set of necessary and sufficient conditions that a matrix commute with its generalized inverse. These conditions involve the existence of certain submatrices and can be expressed using the notion of EPr matrices introduced in 1950 by Schwerdtfeger (10). The main purpose of this paper is to prove the following theorem:

Theorem 2. A necessary and sufficient condition that the generalized inverse of the matrix A (denoted by A+) commute with A is that A+ can be expressed as a polynomial in A with scalar coefficients.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 62 , Issue 4 , October 1966 , pp. 673 - 677
Copyright
Copyright © Cambridge Philosophical Society 1966

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Arghtriade, E.Sur les matrices qui sont pennutables avec leur inverse generalisée. Accad. Naz. dei Lincei (VIII), 35 fasc. 5 (1963), 244–251.Google Scholar
(2)Bjerhammar, A.Application of calculus of matrices to the method of least squares with special reference to geodetic calculations. Trans. Royal Instit. Tech. Stockholm 49 (1951), 1–86.Google Scholar
(3)Bjerhammar, A.Rectangular reciprocal matrices, with special reference to geodetic calculations. Bull. geod. int. (1951), 188–220.CrossRefGoogle Scholar
(4)Jacobson, N.Lectures in abstract algebra, vol. II (New York, 1953).CrossRefGoogle Scholar
(5)Mirsky, L.An introduction to linear algebra (Oxford, 1955).Google Scholar
(6)Moore, E. H.Bull. Amer. Math. Soc. (2) 26 (1920), 394–395.Google Scholar
(7)Moore, E. H.General Analysis, Part 1. Mem. Amer. Philos. Soc. 1 (1935), 197–209.Google Scholar
(8)Penrose, R.A generalized inverse for matrices. Proc. Cambridge Philos. Soc. 51 (1955), 406–413.CrossRefGoogle Scholar
(9)Penrose, R.On best approximate solutions of linear matrix equations. Proc. Cambridge Philos. Soc. 52 (1956), 600–601.CrossRefGoogle Scholar
(10)Schwerdtfeger, H.Introduction to linear algebra and the theory of matrices (Groningen, 1950).Google Scholar