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A Note on Trace-Differentiation and the Ω-operator

Published online by Cambridge University Press:  20 January 2009

A. C. Aitken
A. C. Aitken
Affiliation:
Mathematical Institute, University of Edinburgh.
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In theory of polarizing operators in invariants the operator

where 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.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 10 , Issue 1 , January 1953 , pp. 1 - 4
Copyright
Copyright © Edinburgh Mathematical Society 1953

References

REFERENCES

Aitken, A. C., Proc. Roy. Soc. Edinburgh, 62(1948), 369–377, 374.Google Scholar
Gårding, L., Proc. Edinburyh Math. Soc. (2), 8 (1948), 73–75.CrossRefGoogle Scholar
Turnbull, H. W., Proc. Edinburgh Math. Soc. (2), 1(1927), 111–128; (2), 2(1930), 33–54,1256–264; (2), 8 (1948), 76–86;CrossRefGoogle Scholar
Theory of Determinants, Matrices and Invariants (1928), 72, 114–115, 123.Google Scholar