Let $X$ be an $(N, M)$ random Gaussian matrix where $M<N$. For a given vector $v$, I want to estimate the expectation of:
\begin{align}
E\left[ {{v^T}X{X^T}{v}} \right]
\end{align}
and
\begin{align}
E\left[ {{e^{ - {v^T}X{X^T}{v}}}} \right]
\end{align}



Expectation of exponential of Gaussian random matrix
Decompose $XX^T = O^T \Lambda O$ with $O$ an $M\times M$ orthogonal matrix and $\Lambda={\rm diag}\,(\lambda\_1,\lambda\_2,\ldots \lambda\_M)$ the diagonal matrix of eigenvalues. Define $w=|v|^{-1} Ov$, then
$$v^T XX^T v =|v|^2 \sum\_{m=1}^M \lambda\_m w\_m^2.$$
The matrix $XX^T$ has a [Wishart distribution,](https://en.wikipedia.org/wiki/Wishart_distribution) with independent $O$ and $\Lambda$. It follows that the $w\_m$'s are independent Gaussians with mean zero and variance $1/M$. The probability distribution of the $\lambda\_m$'s is
$$P(\lambda\_1,\lambda\_2,\ldots\lambda\_M)\propto \prod\_{m=1}^M e^{-\lambda\_m/2}\lambda\_m^{(N-M-1)/2}\prod\_{i<j}|\lambda\_i-\lambda\_j|,$$
with $E[\sum\_{m}\lambda\_m]=NM$.


This gives
$$E\left[ {{v^T}X{X^T}{v}} \right]=|v|^2 N.$$
The expectation $E\left[\exp(- {{v^T}X{X^T}{v}}) \right]$ can be evaluated by integration for small $M$,
$$E\left[\exp(- {{v^T}X{X^T}{v}}) \right]=\int\_0^\infty d\lambda\_1\cdots\int\_0^\infty d\lambda\_M \,P(\lambda\_1,\ldots\lambda\_M)\prod\_{m=1}^M(1+2M|v|^2\lambda\_m)^{-1/2},$$
for large $M$ it tends to $e^{-|v|^2 N}$.


