Does this type of matrix expression have a name? $$D'S=CDC^{-1}$$
All the matrices are $n\times n$ and $S$ is a matrix consisting of the inner products of the basis vectors of $D'$, so $D'$ is in general defined with respect to a nonorthonormal basis.
Obviously, the matrix $D'S=E$ is a similar matrix to $D$ and if $S=\mathbf{1}$ (ie $D'$ is in an orthonormal basis) then $D'$ itself is just a similarity transform of $D$, but is there terminology for the general case, something akin to "$D'$ is similar to $D$ with respect to the metric $S$."
As a matter of context, I was trying to answer a question on Chem SE about the relationship between atomic and molecular basis density matrices, when I arrived at an expression similar to the one above. I was hoping to find a way to describe how the bases of $D'$ and $D$ are related, as in the physical context they describe very similar properties. It seems to have something to do with nonorthogonality of the basis of the matrix $D'$.