Could somebody explain to me in simple terms what an isotropic covariance matrix is? I can't find anything online.
A covariance matrix $\mathbf C$ is called isotropic, or spherical, if it is proportionate to the identity matrix: $$\mathbf C = \lambda \mathbf I,$$ i.e. it is diagonal and all elements on the diagonal are equal.
This definition does not depend on the coordinate system; if we rotate coordinate system with an orthogonal rotation matrix $\mathbf V$, then the covariance matrix will transform into $$\mathbf V^\top \mathbf C \mathbf V = \mathbf V^\top \cdot \lambda \mathbf I \cdot\mathbf V = \mathbf V^\top \mathbf V \cdot \lambda \mathbf I = \lambda \mathbf I,$$ i.e. will stay the same.
Intuitively, isotropic covariance matrix corresponds to a "spherical" data cloud. A sphere remains a sphere after rotation.
The covariance is only a function of $|x - x'|$. You can find a definition there.
Edit: sorry I misread, for matrix, the right answer is amoeba's one.