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Matrix Differentiation of the Characteristic Function

Published online by Cambridge University Press:  20 January 2009

H. W. Turnbull
H. W. Turnbull
Affiliation:
St Andrews University.
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The following work is a sequel to three previous communications, and more particularly to the first. The present object is to shew the effect of repeated operation with the matrix differential operator , when it acts upon a scalar matrix formed from an n rowed determinant |xij|, or sums of principal minors, the n2 elements xij being treated as independent variables. Thus when z is a scalar quantity ω z means the matrix [∂z/∂xij], whose ijth element is the derivative.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 2 , Issue 4 , June 1931 , pp. 256 - 264
Copyright
Copyright © Edinburgh Mathematical Society 1931

References

page 256 note 1 I. Turnbull, H. W, On differentiating a matrix, Proc. Edinburgh Math. Soc. (2), 1 (1927), 111–128.CrossRefGoogle Scholar

II. A matrix form of Taylor's Theorem (2), 2 (1929), 33–54.Google Scholar

III. The invariant theory of bilinear forms, Proc. London Math. Soc. (1931).Google Scholar

page 259 note 1 CfDickson, L. E, Modern Algebraic Theories (Chicago, 1926), 48, after replacing P r by C n-l-r.Google Scholar