I have an integral over a subspace of $\mathbb{R}^n \times \mathbb{R}^n$ with an integrand of the form
$$\exp\left(-\frac{1}{2}\left[||u^2|| + \langle u, v \rangle + ||v||^2\right]\right)$$
The subspace is exactly the space for which $u\_{i} = u\_{n-i}$ (assume $n$ is even). In other words, $v$ is a true $n$-dimensional vector, whereas $u$ is two copies of an $n/2$ dimensional vector. 


In order to evaluate this integral, I have been thinking about it as a density over ALL of $\mathbb{R}^n \times \mathbb{R}^n$ of two correlated Gaussians with covariance given by the block matrix
$$\begin{bmatrix}
 A & B \\ B^t & C
\end{bmatrix}
$$
where $A,C$ are $n \times n$ identity matrices and $B$ is the block matrix 
$$\begin{bmatrix}
 I\_{n/2} & I\_{n/2} \\ 0 & 0
\end{bmatrix}
$$
However, given that this is degenerate, I am having trouble finishing the computation. Would greatly appreciate any tips!



Degenerate Gaussian Integral
$\newcommand{\R}{\mathbb{R}}$
Let $U:=\{u\in\R^n\colon u\_i=u\_{n-i}\ \forall i\}$ be your $n/2$-dimensional subspace. I am assuming that your integral is with respect to the product of the Lebesgue measures on $U$ and $\R^n$, and I will denote those measures by $du$ and $dv$, respectively. So, if $\cdot$ denotes the dot product, your integral is 
\begin{align}
 I&:=\int\_U du\,\int\_{\R^n}dv\,\exp\big(-(\|u\|^2+u\cdot v+\|v\|^2)/2\big) \\ 
 &=\int\_U du\,\int\_{\R^n}dv\,\exp\big(-(3\|u\|^2/4+\|v+u/2\|^2)/2\big) \\ 
 &=\int\_U du\,\int\_{\R^n}dw\,\exp\big(-(3\|u\|^2/4+\|w\|^2)/2\big) \\ 
 &=(2\pi)^{n/2}\int\_U du\,\exp\big(-3\|u\|^2/8\big) \\ 
 &=(2\pi)^{n/2}\int\_{\R^{n/2}} dt\,\exp\big(-3\|t\|^2/8\big) \\ 
 &=(2\pi)^{n/2}(2\pi)^{n/4}(4/3)^{n/4}. 
\end{align}
The penultimate equality here holds because both the Euclidean norm and the Lebesgue measure are rotation invariant, whereas the dimension of $U$ is $n/2$; in fact, this is how the Lebesgue measure on $U$ can/should be defined: by the condition that 
\begin{equation}
 \int\_U du\,f(u)=\int\_{\R^{n/2}} dt\,f(Tt) 
\end{equation}
for all nonnegative Borel-measurable functions $f\colon U\to\R$, where $T\colon\R^{n/2}\to U$ is a linear isometry. 


