A Chain Rule for Matrix Functions and Applications
Roy Mathias · SIAM Journal on Matrix Analysis and Applications · 1996
Let f be a not necessarily analytic function and let $A( t )$ be a family of $n \times n$ matrices depending on the parameter t. Conditions for the existence of the first and higher derivatives of f $f( A ( t ) )$ are presented together with formulae that represent these derivatives as a submatrix of $f( B )$, where B is a larger block Toeplitz matrix. This block matrix representation of the first derivative is shown to be useful in the context of condition estimation for matrix functions. The results presented here are slightly stronger than those in the literature and are proved in a considerably simpler way.