I am trying to solve an integral of the form
$I = \int_{\mathcal{S}_{n \times n}(\mathbb{C})} f(X) \; dX$
where $\mathcal{S}_{n \times n}(\mathbb{C})$ is the cone of $n \times n$ positive definite complex Hermitian matrices and
$f : \mathcal{S}_{n \times n}(\mathbb{C}) \to \mathbb{R}$.
I want to apply a one-to-one differentiable change of variable by $Y = g(X)$ with $g : \mathcal{M}_{n \times n}(\mathbb{C}) \to \mathcal{M}_{n \times n}(\mathbb{C})$ and thus I get
$I = \int_{S} |\frac{\partial g(X)}{\partial X}|^{-1} f(g^{-1}(Y)) \; dY$
where $| \cdot |$ is the absolute value of the determinant.
To compute $|\frac{\partial g(X)}{\partial X}|$, I study the eigenvalues and eigenmatrices of the directional derivative
$D_\Delta g(X) = \frac{\partial g(X)}{\partial X}[\Delta] = \lim_{\epsilon \to 0} \frac{g(X + \epsilon \Delta) - g(X)}{\epsilon}$
from the eigenvalues, I can then deduce the determinant.
My problem is that I get a different result depending if I consider $D_\Delta g$ the linear operator on (1) the vector space of complex matrices $\mathcal{M}_{n \times n}(\mathbb{C})$ over the field $\mathbb{C}$, or (2) the vector space of complex Hermitian matrices $\mathcal{H}_{n \times n}(\mathbb{C})$ over the field $\mathbb{R}$ (because the basis of eigen matrices I obtain is different). Note that $\mathcal{H}_{n \times n}(\mathbb{C})$ on the field $\mathbb{C}$ is not a vector space, neither is the cone $\mathcal{S}_{n \times n}(\mathbb{C})$.
I feel like the result I get with (2) is the correct one, but I don't know what are the correct arguments. If so, why should I consider the field $\mathbb{R}$ and not $\mathbb{C}$?
Thanks