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XLVIII.—On Commuting Matrices and Commutative Algebras*

Published online by Cambridge University Press:  14 February 2012

D. E. Rutherford
D. E. Rutherford
Affiliation:
United College, University of St Andrews
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Extract

The structure of commutative associative linear algebras is well known and is usually derived from more general results concerning non-commutative algebras (Cartan, Frobenius). The novelty of the present treatment is that while it avoids the complexities of the non-commutative case, it exhibits the essential relationship between the theory of commuting matrices and that of commutative algebras.

While theorems 1 and 2 of this paper are implicit in the writings of Voss (1889), Taber (1890), and Plemelj (1901), it has been considered worth while to recapitulate these results in the explicit form required for the discussion of commutative algebras. In doing so, some new facts emerge.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 62 , Issue 4 , 1949 , pp. 454 - 459
Copyright
Copyright © Royal Society of Edinburgh 1949

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References

References to Literature

Cartan, E., 1898. “Sur les groupes bilinéaires et les systémes de nombres complexes”, Ann. de Toulouse, XII, 17.Google Scholar
Dickson, L. E., 1930. Linear Algebras, Cambridge.Google Scholar
Frobenius, G., 1903. “Theorie der hypercomplexen Grössen II”, Sitz. Akad. Berlin, 634.Google Scholar
Plemelj, J., 1901. Monatshefte für Math. u. Phys., XII, 82.CrossRefGoogle Scholar
Taber, H., 1890. “On the matrical equation ΦΏ=ΏΦ”, Proc. American Acad. Arts and Sciences, N.S., XVIII, 64.CrossRefGoogle Scholar
Voss, A., 1889. “Ueber die mit einer bilinearen Form vertauschbaren bilinearen Formen”, Sitz. Bayer. Akad. Wiss., XIX, 283.Google Scholar