A good reference to the fact that unitary transforms map normal vectors to normal vectors?

Question

More precisely: given a vector of independent, standard normally distributed random variables, if we apply a unitary transform we obtain again a vector of independent, standard normally distributed random variables.

A good reference?

Answer 1

If $X$ is multivariate normal and $A$ is any real matrix, we can let $Y=AX$ and obtain a multivariate normal $Y$ with

$E[Y]=AE[X]$

and

$\mbox{Cov}(Y)=A \; \mbox{Cov}(X) A^{T}$.

If $X$ is a standard normal ($N(0,I)$), and $A$ is an orthogonal matrix matrix, then $E[X]=0$, so $E[Y]=A0=0$, and $\mbox{Cov}(Y)=AIA^{T}=I$. Thus $Y$ is $N(0,I)$.

This is discussed in many textbooks and references on mathematical statistics. A reasonable starting point is the Wikipedia article on the multivariate normal distribution:

https://en.wikipedia.org/wiki/Multivariate_normal_distribution#Affine_transformation