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 isotropy is an anisotropic distribution.

A characterization of isotropy comes from Vershynin, Lemma 3.2.3: $X\in {\mathbb{R}}^d$ is isotropic iff $\mathbb{E}⟨X,z{⟩}^2=\Vert z{\Vert }_2^2$ for all fixed $z\in {\mathbb{R}}^d$.