How to do change of variables in multivariate normal distribution?

Question

I came across this problem from an exam about recursive estimation I found online:

Let $\newcommand{\R}{\mathbb{R}}$ $x\in\R^n$ and $y\in\R^n$ with a joint Gaussian distribution,

$\begin{bmatrix} x\\ y\\ \end{bmatrix}$ ~ $\mathcal{N}$ $\begin{pmatrix}\begin{bmatrix} \mu_x\\ \mu_y\\ \end{bmatrix}\ ,\begin{bmatrix}\sum_x & \sum_{xy} \\ \sum_{xy}^T & \sum_y \\ \end{bmatrix}\end{pmatrix} $

Find the variance of z = x - y

I couldn't find a change of variable formula for multivariate gaussian distributions online and I really don't know how to possibly solve this. Any help is appreciated

Answer 1

Lets rewrite your variables, $$ q = \begin{bmatrix} x\\ y\\\end{bmatrix}\\ \mu = \begin{bmatrix} \mu_x\\ \mu_y\\\end{bmatrix}\\ \Sigma = \begin{bmatrix} \Sigma_{xx} & \Sigma_{xy}\\ \Sigma_{yx} & \Sigma_{xy} \end{bmatrix} $$ Then what you are referring to is an affine transformation $$ {z} = B q\\ B = \begin{bmatrix} {I} & -{I}\end{bmatrix}\\ $$ According to wikipedia, after an affine transformation, $z$ will be distributed according to a normal distribution $$ z \sim N\left( B\mu, B \Sigma B^T \right). $$