A Note on Trace-Differentiation and the Ω-operator

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Abstract
In theory of polarizing operators in invariants the operatorwhere X = [xij] is an n × n matrix of n2 independent elements xij, 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, 1929, 1931) an exact analogy with results in the ordinary differentiation of the corresponding functions of one scalar variable. Turnbull denotes this operation of trace-differentiation under Ω by Ω8; and we shall follow him. Our purpose is to show how, with a suitably modified Ω, the results may be extended to the case of symmetric matrices X = X′ having ½n(n + 1) independent elements.

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