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This question is similar to this question, but is generalized for rectangular matrices.

Suppose we have matrix $\bf{S} \in \mathbb{R}^{m \times n} $ and vector $\bf{y} \in \mathbb{R}^{n}$.

$\textbf{y}$ is sampled from a multivariate normal distribution with $\Sigma = \sigma^2 I$, where $\mu$ and $\sigma$ is known:

$$ \textbf{y} \sim \mathcal{N}(\mu, \sigma^2 I) $$

$\textbf{S}$ is a fixed, known matrix

What is the variance of the product of $\textbf{Sy}$?

What I have tried so far (attempted working off of same question): \begin{align*} \mathrm{Var}(\textbf{Sy}) &= \mathbb{E}[(\textbf{Sy})^2]-\mathbb{E}[(\textbf{Sy})]^2\\ &= \mathbb{E}[(\textbf{Sy})^T\textbf{Sy}]-\mathbb{E}[\textbf{Sy}]^T\mathbb{E}[\textbf{Sy}]\\ &= \mathbb{E}[\textbf{y}^T\textbf{S}^T\textbf{Sy}] - \mathbb{E}[\textbf{y}^T\textbf{S}^T]\mathbb{E}[\textbf{Sy}] \end{align*}

From here, it seems I would like to remove $\textbf{S}$ from the expected value, but I am not sure how to proceed.

D. Price
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    $Sy$ is a vector, so you cannot proceed like that. The definition is $\operatorname{Var}(Sy)=E\left[(Sy-E(Sy))(Sy-E(Sy))^T\right]$. This reduces to $S\operatorname{Var}(y)S^T$ on simplification. – StubbornAtom Jul 22 '20 at 18:23
  • You may as well assume $\mathbf \mu=\mathbf 0$ and so $\mathbb{E}[\textbf{Sy}]=\mathbf 0$ as it makes no difference to the variances and covariances – Henry Jul 22 '20 at 19:07
  • You then get $\Sigma_\textbf{Sy}=\mathbb{E}[\textbf{S}\textbf{y}\textbf{y}^T\textbf{S}^T] = \textbf{S}\Sigma_\textbf{y}\textbf{S}^T=\sigma^2 \textbf{S}\textbf{S}^T$ – Henry Jul 22 '20 at 19:16

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