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 $S$?

  • 4
    $\begingroup$ Since you mention a paper you found on Google, please link to the paper. $\endgroup$
    – Ben
    Jun 14, 2018 at 4:44
  • $\begingroup$ 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. $\endgroup$
    – ironman
    Jun 14, 2018 at 4:52
  • $\begingroup$ No problem - hopefully my answer helps anyway. $\endgroup$
    – Ben
    Jun 14, 2018 at 5:05
  • $\begingroup$ Suggest to change the title to something a little more precise like "linear transformation of normal gaussian vectors". $\endgroup$
    – JayCe
    Jun 14, 2018 at 13:07

2 Answers 2


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.


For a little bit of visualisation, consider that the Gaussian distribution is scaled by r^2, so multiple independent axes form a Pythagorean relation when scaled by their standard deviations, from which follows that the re-scaled distribution fuzz ball becomes spherical (in n dimensions) and can be rotated about its centre at your convenience.

One of the radial measures is the Mahalanobis distance and is useful in many practical cases where the central limit is applied...


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