derivative of characteristic function of Wishart

Question

is there anybody who can solve this expression: $$\frac{\partial}{\partial_Z} [det(I-2ZA)^{-b/2}]$$ where Z,A are {nxn} symmetric matrices, I is the {nxn} identity matrix and b is a positive scalar. This formula actually coincides with the characteristic function of a Wishart distribution, so I need to know the steps to get to the final result. Thanks a lot!

Answer 1

We assume $Z\in M_n$. The derivative is $Df_Z:H\in M_n\rightarrow -b/2\det(I-2ZA)^{-b/2-1}\det(I-2ZA)tr(-2HA(I-2ZA)^{-1})$

$= b\det(I-2ZA)^{-b/2}tr(A(I-2ZA)^{-1}H)$

and the gradient is $\nabla f(Z)=b\det(I-2ZA)^{-b/2}(I-2A^TZ^T)^{-1}A^T$