Similarity transform with respect to a metric?

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 $$D'$$.

• You have not clearly specified what kind of matrices $C, D, S$ are. Also what is the relationship of $D'$ to $D$, transpose? – Hans Oct 3 '18 at 9:03
• @Hans I've tried to add a little more description of the matrices; if its still unclear, let me know what info about the matrices would help. The relationship of $D'$ and $D$ is what I'm hoping to determine from an answer. The form of my equation is that of a similarity transform $A'=BAB^{-1}$, but with $A'$ split into a matrix in a nonorthogonal basis and an overlap matrix of the inner products of that basis. – Tyberius Oct 3 '18 at 15:33
• So the lhs related to the musical isomorphism? – Emil Oct 3 '18 at 16:22
• @Emil The problem I'm dealing with comes up in the context of quantum chemistry, so I can't say I'm familiar with the concept, but I'd be happy to learn if they are related. – Tyberius Oct 3 '18 at 16:31
• @Tyberius: I wonder if it might be raising or lowering indices that is done on lhs. – Emil Oct 3 '18 at 16:34