Consider a random vector $X\in\mathbb{R}^d$ that is distributed according to a multivariate Gaussian $\mathcal{N}(\mu, \Sigma)$, and the random variable $y = \langle a, X \rangle + e$ for some (fixed) vector $a\in\mathbb{R}^d$ and random variable $e\in\mathbb{R}$ that is independent of $X$ and distributed according to a centered univariate Gaussian $\mathcal{N}(0, \sigma_e^2)$.
It is well-known that $y$ is therefore distributed according as a $\mathcal{N}(a^T \mu, a^T \Sigma a + \sigma_e^2)$.
I am interested in the eigenvalues of the covariance matrix of the random vector $F = \begin{bmatrix} X \\ y \end{bmatrix} \in\mathbb{R}^{d+1}$ which also has a multivariate Gaussian distribution, according to the above.
This covariance matrix (let's call it $Z$) is a block matrix, as follows: $$ Z = \begin{bmatrix} \Sigma & B \\ B^T & \sigma_e^2 + a^T \Sigma a \end{bmatrix} $$ for some appropriate vector $B$ that is the covariance of $X$ and $y$.
If it is not possible to exactly analytically find the largest eigenvalue of $Z$, then I would like to arrive at a (good) upper bound of it. I suspect that it would look something like $\lambda_\text{max}(\Sigma)$ plus something like $\sigma_e^2 + a^T \Sigma a$.