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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#### Best Answer

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: $$f_mathbf{Y}(mathbf{y})=frac{1}{vertmathbf{A}vert}f_mathbf{X}(mathbf{A}^{-1}vertmathbf{y}-mathbf{b}vert)$$

Since $vertmathbf{A}vert$ is constant, $f_{mathbf{Y}}(y)propto f_mathbf{X}(mathbf{A}^{-1}vertmathbf{y}-mathbf{b}vert)$, and since $f_mathbf{X}(mathbf{.})$ is in MV normal form, so is $f_mathbf{Y}(mathbf{y})$.