Inner product between noisy and clean normal vectors

Question

Let $x \in \mathbb{R^n}$ so that $x \sim N(0, \Sigma)$, and let $y = x + e$ so that $e \sim N(0, \sigma^2 I)$.

That is, $y$ is $x$ + Gaussian noise.

Are there known results for $E[ \langle x, y \rangle]$?

Answer 1

Let $b_1, \dots, b_n\in\mathbb{R}^n$ be the basis in which the symmetric matrix $\Sigma$ is diagonal $\mathrm{diag}(\lambda_1, \dots, \lambda_n)$. Let $x_k$, $y_k$ and $e_k$ be the coefficients of $x$, $y$ and $e$ respectively in the basis $b_1, \dots, b_n$

$$ x = x_1b_1 + \dots + x_nb_n\\ y = y_1b_1 + \dots + y_nb_n\\ e = e_1b_1 + \dots + e_nb_n. $$

We have

$$ \begin{align} \mathbb{E}[\langle x, y\rangle] &= \mathbb{E}\left[\sum_{k=1}^nx_ky_k\right] \\ &= \mathbb{E}\left[\sum_{k=1}^nx_k^2 + \sum_{k=1}^nx_ke_k\right] \\ &= \sum_{k=1}^n\mathbb{E}[x_k^2] + \sum_{k=1}^n\mathbb{E}[x_ke_k] \\ &= \sum_{k=1}^n\lambda_k + 0 \\ &= \mathrm{tr}\,\Sigma \end{align} $$

where we assume the noise $e$ is independent of $x$.