Gradient of function of matrix exponential

Question

Suppose I have a differentiable function $\phi: \mathbb{R}^{p\times p} \mapsto \mathbb{R}$ defined as $\phi(\exp(tA))$ where $t$ is a positive scalar and $A$ is a $p\times p$ real matrix. How can I find gradient $\nabla \phi$ with respect to $A$?

Answer 1

Suppose that we have $\psi$ defined by $$ \psi(A) = \phi(\exp(tA)) $$ The chain rule tells us that $$ D_A\psi = [D\phi](\exp(tA)) [D_A \exp(tA)] $$ The derivative of the exponential map may be taken as given here, for example.