Differentiating Matrix Functions

Details Differentiating Matrix Functions Differentiating Matrix Functions

2011-04-02

arxiv.org/abs/1104.0336v1

Mathematics

Functional Analysis

38 pages

Scientific paper

Multivariable, real-valued functions induce matrix-valued functions defined on the space of d-tuples of n-times-n pairwise-commuting self-adjoint matrices. We examine the geometry of this space of matrices and conclude that the best notion of differentiation of these matrix-valued functions is differentiation along curves. We prove that a C^1 real-valued function always induces a C^1 matrix function and give an explicit formula for the derivative. We also show that every real-valued C^m function defined on an open rectangle in the plane induces a matrix-valued function that can be m-times continuously differentiated along C^m curves.

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