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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??

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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 $f_\mathbf{Y}(\mathbf{y})$.

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