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Published online by Cambridge University Press: 15 September 2014
It is well known that the value of a cubic (or odd-dimensional) determinant |aijk| (i, j, k = 1, 2, … n) will reduce identically to zero, if exactly similar rôles are assigned to all the suffixes in determining the signs of the terms. To escape this difficulty, the usual theory of determinants of higher dimensions assigns a special rôle to the first suffix i, and considers that the class of the permutation of the values of i in any term makes no contribution to the sign of the term. This is, in effect, to make the suffix i non-determinantal in character.
page 169 note * The word “determinant” used without any qualification is to be understood to mean “mixed determinant.”
page 169 note † A permutation is of even or odd class, according as it is equivalent to an even or odd number of interchanges. The sign associated with the permutation is + in the former and – in the latter case.
page 169 note ‡ The term “Scott function” is used by Hedrick, Annals of Math., 2nd series, 1–2, to denote a “signless determinant” |aij|.
page 169 note § The question of the validity of this theorem for n = 1 is one of some interest, though it has little or no practical importance. While, according to this theorem, the pure one-element determinant |a| vanishes, we are assured with equal certainty from the theory of elimination from linear equations, that the same pure determinant should have the value a. The contradiction arises from the fact that our theory of determinants is concerned only with determinants of definite dimensionality. A determinant of order one is of indeterminate dimensions, corresponding to the geometrical fact that a point could be considered as the shrinking limit of a piece of continuum of any number of dimensions. Hence if we are given a one-element determinant |a|, and if we are told in addition that it is a p-dimensional determinant with q Scott suffixes, we would logically be in a position to evaluate it ; namely |a| = a or 0, according as p – q is even or odd.
The following slightly different point of view may perhaps be considered preferable. The concept of determinant as a number associated with an ordered spatial array, is based essentially on the notion of suffix. Now a suffix is something which possesses necessarily a field of variation. But if the order n is equal to one, there is no field of variation and therefore, strictly speaking, there are no suffixes (unless convention steps in and says that there are). Hence determinants of order unity are determinants of zero dimensions, and therefore may be legitimately considered to be meaningless.
As this case does not occur in practice, it is immaterial which of these two views we adopt.
page 173 note * This remarkable theorem and its proof are due in substance to Campbell, Proc. L.M.S., vol. xxiv, 1st series.
page 174 note * This proof is easily seen to be also valid for the case in which some or all of n 1n 2 … nr are equal to 1 or 2.
page 179 note * For the invariant theory of the multilinear form see Study, Vectorenrechnung (Braunschweig, 1923), p. 110. For apolar theory of double-binary form see Kasner, Trans. Am. Math. Soc., vol. i.
page 182 note * For a few more examples of what we have termed extensional invariants, see Hedrick, loc. cit.
