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 $D'$.

  • $\begingroup$ You have not clearly specified what kind of matrices $C, D, S$ are. Also what is the relationship of $D'$ to $D$, transpose? $\endgroup$ – Hans Oct 3 '18 at 9:03
  • $\begingroup$ @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. $\endgroup$ – Tyberius Oct 3 '18 at 15:33
  • $\begingroup$ So the lhs related to the musical isomorphism? $\endgroup$ – Emil Oct 3 '18 at 16:22
  • $\begingroup$ @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. $\endgroup$ – Tyberius Oct 3 '18 at 16:31
  • $\begingroup$ @Tyberius: I wonder if it might be raising or lowering indices that is done on lhs. $\endgroup$ – Emil Oct 3 '18 at 16:34

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