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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A symmetry property of the Fréchet derivative


Author: Roy Mathias
Journal: Proc. Amer. Math. Soc. 120 (1994), 1067-1070
MSC: Primary 15A04; Secondary 15A24, 15A57, 15A69
MathSciNet review: 1216819
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Abstract: Let $ A$ and $ B$ be $ n \times n$ matrices. We show that the matrix representing the linear transformation

$\displaystyle X \mapsto {(AXB + BXA)^T}$

(which is from the space of $ n \times n$ matrices to itself) with respect to the usual basis is symmetric and show a similar symmetry property for the Fréchet derivative of a function $ f(X) = \sum\nolimits_{i = 0}^\infty {{a_i}{X^i}} $ defined on the space of $ n \times n$ matrices.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1994-1216819-0
PII: S 0002-9939(1994)1216819-0
Keywords: Kronecker product, tensor product, Fréchet derivative, complex symmetric matrix
Article copyright: © Copyright 1994 American Mathematical Society