# Solved – The linear transformation of the normal gaussian vectors

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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