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III.—The Invariant Theory of the Correlation

Published online by Cambridge University Press:  15 September 2014

H. W. Turnbull
H. W. Turnbull
Affiliation:
University of St Andrews
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Extract

Beyond the ternary and quaternary cases considered by Clebsch (1891) and Weitzenböck (1910), there appears to be no writing on the bilinear form in cogredient variables—the correlation—from the point of view of the projective invariant theory. In the present work a complete system of concomitants is found for the single bilinear ground form. Incidentally it is shown that such a system belongs not to the single form but to the simultaneous system of two mutually conjugate forms. Symbolic methods are used, and the processes are analogous to those employed in the cases of two quadratic forms (Turnbull and Williamson, 1929), and also of one bilinear form (Turnbull, 1932) in contragredient variables. In contrast with these previous results it is remarkable that the present system involves no forms of weight greater than two (§ 3, § 6).

Type
Proceedings
Information
Proceedings of the Royal Society of Edinburgh , Volume 55 , 1936 , pp. 27 - 41
Copyright
Copyright © Royal Society of Edinburgh 1936

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References

References to Literature

Clebsch, Lindemann, 1891. Vorlesungen über Geometrie (Leipzig), Bd. ii (1), P. 343.CrossRefGoogle Scholar
Turnbull, H. W., 1927. “The Invariant Theory of the Quaternary Quadratic Complex,” Proc. Roy. Soc. Edin., vol. xlviii, pp. 70–91.Google Scholar
Turnbull, H. W., 1929. Theory of Determinants, Matrices and Invariants, Glasgow.Google Scholar
Turnbull, H. W., 1930. “The Complete System of two Quadratic Forms in n Variables,” Proc. London Math. Soc. (2), vol. xxx, pp. 469–480.CrossRefGoogle Scholar
Turnbull, H. W., 1932. “The Invariant Theory of a General Bilinear Form,” Proc. London Math. Soc., vol. xxxiii, pp. 1–21.CrossRefGoogle Scholar
Turnbull, H. W., and Williamson, J., 1929. “Further Invariant Theory of two Quadratics in n Variables,” Proc. Roy. Soc. Edin., vol. 1, pp. 8–25.Google Scholar
Weitzenböck, R., 1910. “Das Formensystem der Korrelation im R3,” Rend, di Palermo, vol. xxx, pp. 122–135.CrossRefGoogle Scholar