gaussian random vectors are closed under linear transformations

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Theorem

Let $x \in R^{d} \sim (μ, Σ)$ be a gaussian random vector, $A \in R^{d^{'} \times d}$ , and $b \in R^{d^{'}}$ . Then $A x + b$ is also a gaussian random vector with

Law (A x + b) = N (A μ + b, A Σ A^{T})

ie, gaussian random vectors are closed under linear transformations.

File	Last Modified
Random Matrix Lecture 01	2025-09-11
standard gaussian random vectors are orthogonally invariant	2025-09-05

gaussian random vectors are closed under linear transformations

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