# constant matrix times Multivariate Gaussian distribution?

Suppose I have a multivariate Gaussian distribution x and a constant matrix A. I know how to calculate the mean and covariance of Ax but how can I prove that Ax will also be multivariate gaussian??

• @Gue : I know how to find both mean and co variance of the new distribution, but how can I show that new distribution will also be gaussian Jun 28, 2019 at 9:17
• An easy way to prove this is by using characteristic functions, see e.g. math.stackexchange.com/questions/605816/… Jun 28, 2019 at 9:57
• There are myriad ways to establish this: see our list of characterizations of Gaussian distributions for some ideas.
– whuber
Jun 28, 2019 at 14:10

In addition to characteristic functions, one can use Jacobians to find an expression for RV transformations, i.e. when we set $$\mathbf{Y}=\mathbf{AX}+\mathbf{b}$$, it's further simplified to: $$f_\mathbf{Y}(\mathbf{y})=\frac{1}{\vert\mathbf{A}\vert}f_\mathbf{X}(\mathbf{A}^{-1}\vert\mathbf{y}-\mathbf{b}\vert)$$
Since $$\vert\mathbf{A}\vert$$ is constant, $$f_{\mathbf{Y}}(y)\propto f_\mathbf{X}(\mathbf{A}^{-1}\vert\mathbf{y}-\mathbf{b}\vert)$$, and since $$f_\mathbf{X}(\mathbf{.})$$ is in MV normal form, so is $$f_\mathbf{Y}(\mathbf{y})$$.