I am facing difficulty in proving the following statement. It is given in a research paper found on Google. I need help in proving this statement!

Let $X= AS$, where $A$ is orthogonal matrix and $S$ is gaussian. The

isotopic behavior of the Gaussian $S$ which has the same distribution in

any orthonormal basis.

How is $X$ Gaussian after applying $A$ on $S$?

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#### Best Answer

Since you have not linked to the paper, I don't know the context of this quote. However, it is a well-known property of the normal distribution that linear transformations of normal random vectors are normal random vectors. If $boldsymbol{S} sim text{N}(boldsymbol{mu}, boldsymbol{Sigma})$ then it can be shown that $boldsymbol{A} boldsymbol{S} sim text{N}(boldsymbol{A} boldsymbol{mu}, boldsymbol{A} boldsymbol{Sigma} boldsymbol{A}^text{T})$. Formal proof of this result can be undertaken quite easily using characteristic functions.

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