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The projective invariants of four mediale

Published online by Cambridge University Press:  20 January 2009

H. W. Turnbull
H. W. Turnbull
Affiliation:
The University, St Andrews.
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The theory of four particular linear forms, or matrices of k columns and 2k rows, occurred to me many years ago in an attempt to study the invariants of any number of compound linear forms, or subspaces within a space of n dimensions. In what follows, the invariant theory is given, and its significance for a study of the general matrix of k rows and columns is suggested. The collineation used in §4 was considered by Mr J. H. Grace, who emphasized the importance of the k cross ratios upon transversal lines of four [k−1]'s in [2k−1]. It seemed appropriate to examine these cross ratios which are irrational invariants μi, of the figure of four such spaces, and to work out their relation to the known rational invariants Xi. The main result is given in § 5 (7). In § 5 (10) it is shewn that the harmonic section of a line transversal of the four spaces exists when a linear relation holds between the invariants.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 7 , Issue 1 , October 1942 , pp. 55 - 72
Copyright
Copyright © Edinburgh Mathematical Society 1942

References

page 55 note 1 “Extension of a geometrical porism and other theorems,” Proc. Camb. Phil. Soc., 25 (1929), 421–432.CrossRefGoogle Scholar

page 55 note 2 Phil. Trans. Royal Soc., A 239 (1942), 233–267 (No. 805).CrossRefGoogle Scholar

page 56 note 1 Turnbull, , Theory of Determinants, Matrices and Invariants (Blackie, 1928), 203.Google Scholar

page 56 note 2 The symbol A is convolved explicitly when all its k partial symbols a occur in one bracket factor (e 1……e 2k), and implicitly when they occur distributed over two or more such factors. In the latter case the expression involves several terms T, the sum of which is necessarily unaltered by determinantal permutation of the partial symbols a. (cf. loc. cit. p. 46).

page 60 note 1 This matrix, P of triangular form with binomial coefficients as elements, was studied in the Journal London Math. Soc., 2 (1927), 242–4.Google ScholarCfVaidyanathaswamy, , “Integer roots of the unit matrix,” Jovmal London Math. Soc., 3 (1928), 121–4.CrossRefGoogle Scholar

page 61 note 1 CfGrassmann, , Ausdehnungslehre (1862), § 113 p. 83.Google ScholarForder, , Calculus of Extension (Cambridge 1941), 217–249.Google Scholar

page 71 note 1 CfTurnbull, and Aitken, , Canonical Matrices (Blackie, 1932), p. 48.Google Scholar

page 71 note 2 Mehrdimensionale Räume, Encyk. Math. Wissenschaften, III. C 7, 841, where an earlier result by Segre (Math. Annalen, 24 (1884), 152–156CrossRefGoogle Scholar), is utilized, which concerns two quadrics. This is relevant since any collineation can be resolved into successive reciprocation in two quadrics by a theorem of Frobenius.