On notation of $n$-dimensional Gaussian distribution

Question

could anyone please help me with the following question?

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?

Thanks in advance!

Answer 1

I've never heard it being called an isotropic covariance matrix, but that doesn't necessarily mean much. But since it's a Gaussian distribution, if the covariance matrix is diagonal then the variables are all independent. So then you can say, for example, that

The elements of $\mathbf{y}$ are independent with means $\boldsymbol{\mu}$ and equal variance $\sigma^2$.