isotropic distributions

Intuitively, an isotropic distribution is rotationally invariant. That is, it is the same in all directions when viewed from the origin.

For instance, a normal $N(\mu ,{\sigma }^{2}I)$ is isotropic, because the variance is the same in all directions.

Mathematically, we say a distribution $P$ is isotropic if for all orthogonal matrices $Q$,

$$X=QX,$$

in distribution, where $X\sim P$. The opposite of anisotropy is an isotropic distribution.