Reference for gradient expression of a function on matrices

Question

I'm looking for a reference (I suppose the statement is correct) for the following formula:

$$ \langle\nabla f(\rho)^\dagger,V\rangle=\left.\frac d{dt} f(\rho+tV)\right|_{t=0} $$

for any direction $V \in \mathbb{C}^{n \times n}$.

For a differentiable function $f : \mathbb{C}^{n \times n} \to \mathbb{C}$ and a matrix $\rho = (\rho_{ij})_{ij} \in \mathbb{C}^{n \times n}$ we define the gradient of $f$ in $\rho$ as

$$ \nabla f(\rho)=\Big(\frac{d}{d\rho_{ij}} f(\rho)\Big)_{ij}. $$

I used the trace inner product

$$ \langle A,B\rangle = \text{tr} \left(A^\dagger B\right). $$

Please tell me, if I'm wrong or give me a book, that I can cite in my thesis.

Answer 1

I cannot cite a reference offhand, but it's easy to prove.

Let $G(X)=\nabla f(X)$ and assume that $X$ is itself a function of $t$ $$X(t)=\rho +tV$$

Then the differential can be expressed in terms of the Frobenius product as $$\eqalign{ df &= G^* : dX \cr &= G^* : V\,dt \cr }$$ So the derivative of $f$ with respect to $t$ is $$\eqalign{ \frac{df}{dt} &= G^* : V \cr &= {\rm tr}(G^\dagger V) \cr }$$