# Complex matrix derivative

Let $$\mathbf{V}\in \mathbb{C}^{N \times N}$$ be an Hermitian and positive definite matrix. Let $$\mathrm{vec(\mathbf{V})} \in \mathbb{C}^{N^2}$$ be the classical vectorization operator. Let $$|\mathbf{V}|$$ be the determinant of $$\mathbf{V}$$. How can I evaluate the following complex derivative? $$$$\frac{\partial|\mathbf{V}|^{1/N}}{\partial\mathrm{vec(\mathbf{V})}^T}$$$$

Thanks!

Taking a derivative with respect to $$\operatorname{vec}(V)^T$$ will just give us a rearranged version of the derivative with respect to $$V$$.
For the derivative with respect to $$V$$, we could use the chain rule along with the matrix calculus result $$\frac{\partial |V|}{\partial V} = \operatorname{adj}(V) = |V|\cdot V^{-1},$$ where adj denotes the adjugate matrix. From there, $$\frac{\partial |V|^{1/N}}{\partial V} = \frac 1N \cdot |V|^{(1-N)/N} \cdot \frac{\partial |V|}{\partial V} = \frac{|V|^{1/N}}{N}\cdot V^{-1}.$$ It now suffices to rearrange the entries of this matrix, according to however the derivative of a function with respect to a row-vector is meant to be arranged.
• Thanks for the answer! This is fine with me, but my concern was about the fact that $\mathbf{V}$ is a complex Hermitian matrix. I’m not sure about the need to use the “Wirtiger calculus” here.. – Vuk Feb 7 '20 at 15:33