Compute integral containing a matrix

Question

Let $\mathbf{H}= \begin{pmatrix} h_{11} & h_{12} \\ h_{21} & h_{22} \end{pmatrix}$ and $P(\mathbf{H}$) the joint probability distribution of $\mathbf{H}$ given by: $e^{-(a+ \sum\limits_{i=1}^{2}\sum\limits_{j=1}^{2} b_{ij}|h_{ij}|^2 ) }$. How to compute the following integrals: \begin{equation} \int P(\mathbf{H}) d\mathbf{H}=1 \end{equation} \begin{equation} \int P(\mathbf{H})|h_{ij}|^2 d\mathbf{H}=\sigma^2 \end{equation} in order to calculate the coefficients $a$ and $b_{ij} (\forall i$ and $j)$.

Answer 1

For the case where it is possible to diagonalize the Matrix in the Exponent this would be a good idea. As the Jacobian is Unity, you end up with ordinary Gaussian Integrals over the Eigenvalues for $(1)$. $(2)$ then can be generated by differentiation of (1) with respect to $b_{ij}$