isotropic gaussian

Isotropic gaussian

An isotropic gaussian is an multivariate distribution where the covariance matrix $\Sigma$ is diagonal and has the same variance across all dimensions:

$$\Sigma ={\sigma }^{2}I$$

Here, $I$ is the identity matrix and ${\sigma }^2$ is the variance.
Since the variance is the same everywhere, the distribution is circular / spherical.
It is a special case of a factored gaussian with fixed varriance:

$$\mathcal{N}(\mu ,\Sigma )=\prod _{i=1}^n\mathcal{N}({\mu }_i,{\sigma }^2)$$

Likewise the dimensions / variables are independent.
See isotropic distribution for general details.