There are component of a cost function $f(\mathbf{s})$ where $\mathbf{s}\in\mathbb{C}^N$ is the parameter (input) of the function and $\mathbb{C}$ denotes the complex dimension. There are matrices $\mathbf{A}\in\mathbb{C}^{N\times N}$ which is constant, $\mathbf{B}(\mathbf{s})\in\mathbb{C}^{N\times N}$ which is a matrix such that $$\mathbf{B}(\mathbf{s})=\mathbf{C}\mathbf{s}\mathbf{s}^H\mathbf{C}^H+\mathbf{I}$$ where $\mathbf{C}\in\mathbb{C}^{N\times N}$ is a constant matrix.
Notation: $\mathbf{s}^*$ is a complex conjugate of $\mathbf{s}$. $\mathbf{s}^H$ is a Hermitian of $\mathbf{s}$, which is a transpose of complex conjugate of $\mathbf{s}$.
How can I derive the Wirtinger complex gradient of a cost function, i.e., $\nabla_{\mathbf{s}^*}f(\mathbf{s})$?
The tricky part is $\mathbf{B}^{-1}(\mathbf{s})$. I found following for the gradient of a inverse matrix, which might be helpful.: Derivative of the inverse of a matrix
