# Gradient of function of matrix exponential

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 $A$?

• As far as I know the exponental of a matrix id defined only for square martices. Commented Aug 24, 2015 at 19:38
• @Taha can we assume that $p=q$? Commented Aug 24, 2015 at 19:38
• I fixed the typo. Now p = q.
– Taha
Commented Aug 24, 2015 at 19:59

## 1 Answer

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.

• Thank you for the answer. Is it possible to point out a simpler article for derivative of just matrix exponential? I do not have the proper background in group theory and it takes some time to fully understand that article.
– Taha
Commented Aug 25, 2015 at 1:04
• You might prefer this, or perhaps something here might help. It isn't going to be a nice function, at any rate. Commented Aug 25, 2015 at 1:16
• Thanks. Do you mean the answer will be far from being used in a practical gradient descent optimization algorithm?
– Taha
Commented Aug 25, 2015 at 1:25
• @Taha actually, this might work well numerically. One expression is a rapidly converging infinite sum, and the other is an integral. It might be easy to get a good approximation... do you want me to clarify that infinite sum? Commented Aug 25, 2015 at 13:24
• Thank you. I also found a couple of papers on numerical computation. After I read them, I will supplement your answer with explicit formulas for future reference of people like me who are just looking for explicit solutions :)
– Taha
Commented Aug 25, 2015 at 15:34