I have been thinking about a problem and I have an intuition about it but I don’t seem to know how to properly address it mathematically, so I’m sharing it with you hoping to get help. Suppose I have two $ntimes n$ real matrices $C$ and $M$ and consider the Gaussian integral:

$$I = Nint e^{-frac{1}{2}ilangle x, C^{-1} xrangle} e^{langle x, M xrangle}dx$$

where $N$ is a normalizig constant and I’m writting:

$$langle x, A x rangle = sum_{i,j}x_{i}A_{ij}x_{j}$$

the inner product of $x$ and $Ax$ on $mathbb{R}^{n}$. $C$ is the covariance of the Gaussian measure; moreover, suppose $M$ is not invertible and has $1 le k < n$ linearly independent eigenvectors associated to the eigenvalue $lambda = 0$. All other eigenvectors of $M$ are also linearly independent, but associated to different nonzero eigenvalues.

This is my problem. I’d like to know how does the formula for the Gaussian integral $I$ changes if I was to integrate over the subspace $S$ spanned by the eigenvectors $v_{1},…,v_{k}$ associated to $lambda = 0$. Intuitively, this integral wouldn’t have the $e^{langle x, Mx rangle}$ factor because $Mequiv 0$ in this subspace. In addition, since $S$ is a $k$-dimensional subspace, I’d expect this integral would become some sort of Gaussian integral over a new variable $y$ which has now $k$ entries.

I would like to know if my intuition is correct and if it is possible to explicitly write this new integral over $S$, which I was not able to do by myself. Thanks!