On notation of $n$-dimensional Gaussian distribution

Let $\Sigma$ be the covariance matrix of an $n$-dimesnional Gaussian distribution. The generic requirement for $\Sigma$ is to be symmetric and positive definite. However, what's happening when $\Sigma$ is diagonal with equal entries, i.e., $\Sigma=\text{diag}\{\sigma^2,\dots,\sigma^2\}$. I mean, how is it called? Is the "isotropic covariance matrix" a proper name for such a covariance matrix? If so, is the Gaussian distribution called an "isotropic Gaussian distribution"? If not, then what? Anyway, do the "isotropic {Gaussian distribution, covariance matrix}" exist?
The elements of $\mathbf{y}$ are independent with means $\boldsymbol{\mu}$ and equal variance $\sigma^2$.