Differentiating matrix functions
Kelly Bickel · Operators and Matrices · 2013
Real-valued functions on R d induce matrix-valued functions defined on the space of d -tuples of n n pairwise-commuting self-adjoint matrices. We examine the geometry of this space of matrices and conclude that a suitable notation of differentiation of these matrix functions is differentiation along curves. We prove that continuously differentiable real-valued functions induce continuously differentiable matrix functions and give a formula for the derivative. We also show that real-valued m -times continuously differentiable functions defined on open rectangles in R 2 induce matrix functions that can be m -times continuously differentiated along m -times continuously differentiable curves.