Let $X$ be a multivariate normal random variable with mean $\boldsymbol{\mu}$ and variance matrix $\mathrm{\Sigma}$. Next, define


Suppose that $Y = AX$ where $A$ is appropriate matrix. Can we say that the distribution of $Y$ is same as $X$ if and only if $A$ is an orthogonal symmetric matrix?


Thank you Iosif for pointing out the mistake. I modified the question.



Multivariate normal distribution and orthonormal transformation
$\newcommand\Si\Sigma$First here, there is no such thing as an orthonormal matrix. So, let us assume that you meant an orthogonal matrix instead.


Then we have this question:



> 
> Suppose that $X\sim N(\mu,\Si)$ and $Y = AX$, where $A$ is appropriate matrix. Can we say that the distribution of $Y$ is same as $X$ if and only if $A$ is an orthogonal matrix?
> 
> 
> 


The answer to this is that neither the "if" part of this conjecture nor its "only if" holds in general.


Indeed, the distribution of $Y=AX$ is $N(A\mu,A\Si A^\top)$. So, the distribution of $Y$ is same as $X$ if and only if
$$A\mu=\mu \tag{1}\label{1}$$
and
$$A\Si A^\top=\Si. \tag{2}\label{2}$$


If $\Si$ is nonsingular, then condition \eqref{2} can be rewritten as $A\_\Si A\_\Si ^\top=I$, where
$$A\_\Si:=\Si^{-1/2}A\Si^{1/2}.$$
So, in this case, the distribution of $Y$ is same as $X$ if and only if \eqref{1} holds and $A\_\Si$ is an orthogonal matrix.


