Gaussian distribution is isotropic?

I was in a seminar today and the lecturer said that the gaussian distribution is isotropic. What does it mean for a distribution to be isotropic? It seems like he is using this property for the pseudo-independence of vectors where each entry is sampled from the normal distribution.

TLDR: An isotropic gaussian is one where the covariance matrix is represented by the simplified matrix $\Sigma = \sigma^{2}I$.

Some motivations:

where $\mu$ is the mean and $\Sigma$ is the covariance matrix.

Consider how the number of free parameters in this Gaussian grows as the number of dimensions grows.

$\mu$ will have a linear growth.
$\Sigma$ will have a quadratic growth!

This quadratic growth can be very computationally expensive, so $\Sigma$ is often restricted as $\Sigma = \sigma^{2}I$ where $\sigma^{2}I$ is a scalar variance multiplied by an identity matrix.

Note that this results in $\Sigma$ where all dimensions are independent and where the variance of each dimension is the same. So the gaussian will be circular/spherical.

Disclaimer: Not a mathematician, and I only just learned about this so may be missing some things 🙂

Hope that helps!