Functions of linear operators: Parameter differentiation (Q2738007)

The authors derive a useful expression for the matrix elements \([\partial f[A(t)]/\partial t]_{ij}\) of the derivative of a function \(f[A(t)]\) of a diagonalizable linear operator \(A(t)\) with respect to the parameter \(t\) at \(t_0\). The function \(f[A(t)]\) is supposed to be an operator acting on the same space as the operator \(A(t)\) which is assumed to have a nondegenerate, pure point spectrum.