I have the following integral I wish to compute: \begin{equation} Z=\int\left( \prod_i \mathrm{d}\phi_i \right)\exp\left\{-\frac{1}{2} \sum_{i, j} J_{i j} \cos(\phi_{i}) \cos(\phi_{j})-h \sum_{i} \cos(\phi_{i})\right\} \end{equation}
I know all the eigenvalues of the matrix $\mathbf{J}$: Am I allowed to replace $J_{ij}$ by its eigenvalues $\lambda_j$?
Can I express the integral $Z$ as follows? \begin{equation} Z=\int\left( \prod_i \mathrm{d}\phi_i \right)\exp\left\{-\frac{1}{2} \sum_{i, j} \lambda_j \cos(\phi_{i}) \cos(\phi_{j})-h \sum_{i} \cos(\phi_{i})\right\} \end{equation}
Is this correct? Or do I need to add some constant in front of the integral, or does it get more complicated than that?