Let $M=\left (\omega\mathbb{I}-A\right )\left(\omega^{*}\mathbb{I}-A^{\dagger}\right)$ be a Hermitian matrix of size $n\times n$ where $A$ is a real non symmetric matrix and $\omega=a+\mathrm{i}b$. $A^{\dagger}$ represents the conjugate transpose of $A$.
I want to compute $\det[M]^{-\frac{1}{2}}$.
I know that for a real symmetric matrix $\Sigma$ we can represent its determinant as a gaussian integral with real variables $x_i$: $$ \frac{1}{|\Sigma|^{1 / 2}}=\int \frac{1}{(2 \pi)^{n / 2}} \exp \left(-\frac{1}{2}\mathbf{x}^{T} \Sigma\mathbf{x}\right)\mathrm{d}\mathbf{x}.$$
However in my case $M$ has complex values. I was wondering if we could extend this integral representation to Hermitian matrices. Among the feedback I got, these are the candidates: \begin{equation} \det[M]^{-\frac{1}{2}}=\int \left ( \prod_{i} \frac{\mathrm{d} x_i}{\sqrt{2 \pi / i}}\right ) \exp \left\{-\frac{\mathrm{i}}{2} \sum_{i j }x_i\left (\sum_k\left(\omega \delta_{i k}-A_{i k}\right)\left(\omega^* \delta_{k j}-A_{k j}^T\right)\right ) x_j\right\}. \end{equation} \begin{equation} \det[M]^{-\frac{1}{2}}=\int\left(\prod_i \frac{d^{2} z_{i}}{\pi}\right) \exp \left\{-\sum_{i, j, k} z_{i}^{*}\left(\omega^{*} \delta_{i k}-J_{i k}^{T}\right)\left(\omega \delta_{k j}-J_{k j}\right) z_{j}\right\} \end{equation} The second one involving complex variables seems intuitively the best suited. However I do not know whether this is correct, and I could use a simpler integral then I would prefer very much so.
Why would not this work: $$ \det[M]^{-\frac{1}{2}}=\int \left ( \prod_{i} \frac{\mathrm{d} x_i}{\sqrt{2 \pi }}\right ) \exp \left\{-\frac{1}{2} \sum_{i j }x_i\left (\sum_k\left(\omega \delta_{i k}-A_{i k}\right)\left(\omega^* \delta_{k j}-A_{k j}^T\right)\right ) x_j\right\}. $$
I am very curious on what the correct way would be. Any remark or advice would be greatly appreciated!
edit: I consider the case where $A$ is real, and does not have complex entries anymore.
Second edit: I was told that I had to integrate over complex $z_i$ rather than real $x_i$. If this is true I would like to know why I can't use real integration.