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 orthonormal matrix?

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.