A Note on Trace-Differentiation and the Ω-operator
Proceedings of the Edinburgh Mathematical Society
1. In theory of polarizing operators in invariants the operator where X -[x$ is an n x n matrix of n 2 independent elements x^, holds an important place. Acting upon particular scalar functions of X, namely the spur or trace of powers of X, or of polynomials or rational functions of X with scalar coefficients, it exhibits (Turnbull,. 1927 (Turnbull,. , 1929(Turnbull,. , 1931 an exact analogy with results in the ordinary differentiation of the corresponding functions of one scalar variable.
... alar variable. Turnbull denotes this operation of trace-differentiation under Q by Q 8 ; and we shall follow him. Our purpose is to show how, with a suitably modified Q, the results may be extended to the case of symmetric matrices X -X' having \n(n + 1) independent elements. Such extensions forced themselves on the notice of the author in some work (Aitken, 1946) on the estimation of statistical parameters. At the same time and independently Garding was using the determinant of the same modified Q. to obtain the analogue (Garding, 1947) of Cayley's determinantal theorem (cf. Turnbull, 1928, p. 114) namely | Q | | I | f = r ( r + | ) ( r + l ) . . . ( r + i » -| ) | I | ' -1 , where in (3) the matrix X is symmetric and O is modified. 2. We first establish the result 1 (2), following one of Turnbull's proofs, when r is a positive integer. The extension to the case of negative powers of X and to rational functions will then be taken in hand. The corresponding results for a symmetric matrix X follow without difficulty. It is of advantage to use tensor notation and the summation convention. We denote x t j by x. , and we agree that the summation convention shall hold for repeated Greek indices, but not for italic.