Given $v$ a random Gaussian distributed vector, is it possible to prove that $r = Rv$ is also Gaussian distributed? Any formula-link-information on how to start working on it is highly appreciated.
Thanks.
2026-03-25 15:38:57.1774453137
Is a random Gaussian distributed vector rotational invariant?
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HINT: The sum of Gaussian variables/vectors (not necessarily independent) is indeed Gaussian.
Hence, for each component of vector $r$ we can write $$r_i = (Rv)_i = \sum_{j} R_{ij} v_j,$$ which is indeed Gaussian, given that $v_j$ is Gaussian.