# 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 $S$?

• Since you mention a paper you found on Google, please link to the paper. – Ben Jun 14 '18 at 4:44
• Sorry I search in Private mode and now I am not able to track it. Actually it is related to Independent Component analysis in unsupervised learning. – ironman Jun 14 '18 at 4:52
• No problem - hopefully my answer helps anyway. – Ben Jun 14 '18 at 5:05
• Suggest to change the title to something a little more precise like "linear transformation of normal gaussian vectors". – JayCe Jun 14 '18 at 13:07

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.