Representing the determinant of a Hermitian matrix as an integral

Question

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.

Answer 1

The following 2 Gaussian integral representations seem relevant to OP's problem:

1. Given a symmetric complex matrix $A\in{\rm Mat}_{n\times n}(\mathbb{C})$

   • (i) such that the matrix ${\rm Re}A$ is positive definite,

   • (ii) or such that $A$ is an invertible imaginary matrix,

   then the Gaussian integral is well-defined and is given by $$ \int_{\mathbb{R}^n} \! d^n x ~e^{-\frac{1}{2} x^T A x} ~=~ \sqrt{\frac{(2\pi)^n}{\det A}}.\tag{1}$$ See e.g. this related Phys.SE post.

2. Given an invertible real matrix $A\in{\rm Mat}_{n\times n}(\mathbb{R})$, then $$ \int_{\mathbb{R}^{2n}} \! d^n x~d^ny ~e^{ix^T A y} ~=~ \frac{(2\pi)^n}{|\det A|}.\tag{2}$$

Case (2) reduces to the case (1.ii) if we build the symmetric imaginary matrix $\begin{pmatrix} 0 & -iA \cr -iA^T & 0 \end{pmatrix}$ of twice size.