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The commuting inverses of a square matrix

  • M. J. Englefield (a1)

An inverse AI for an arbitrary matrix A was first given by Moore (4). Since the application to solution of linear equations only depended on the property A AI A = A, Bjerhammar (2) used this equation to define the set of generalized inverses AI. If A is regular, then only the regular inverse A−1 satisfies this definition. If A is a generalized inverse of AI, so that AI = AI AAI, then AI is a reciprocal inverse.

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(1)Afriat, S. N.Proc. Cambridge Philos. Soc. 55 (1959), 51.
(2)Bjerhammar, A.A generalized matrix algebra. Kungl. Tekniska Hogskolans Handlingar, no. 124 (1958).
(3)Giorgi, G.Atti Accad. Naz. Lincei, VI 8 (1928), 3. Or see MacDuffee, The theory of matrices, p. 100 (New York, 1946).
(4)Moore, E. M.Bull. Amer. Math. Soc. 26 (1920), 394.
(5)Penrose, R.Proc. Cambridge Philos. Soc. 51 (1955), 406.
(6)Penrose, R.Proc. Cambridge Philos. Soc. 52 (1956), 18.
(7)Wedderburn, J. H. M.Lectures on matrices. (Amer. Math. Soc. Colloquium Publications, vol. XVII, 1934), pp. 2931.
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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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