Consider the square matrices over a (local) ring $R$, up to conjugation, $A\rightarrow UAU^{-1}$, where $U$ is an invertible matrix over $R$. Such an equivalence embeds into the "left-right" equivalence, $A\rightarrow UAV$, by the standard trick: $A=UBU^{-1}$ iff $t_0A+t_1 I=U(t_0 B+t_1 I)V$. Here $t_0,t_1$ are some new formal variables, $I$ is the identity matrix.
(This goes in the style: one can embed the representation of any quiver to the Kronecker quiver.)
Can one "embed" by some similar trick the congruence, $A\rightarrow UAU^T$, into the left-right equivalence? (i.e. $A=UBU^T$ iff for some associated matrices $\mathcal{A,B}$ and some invertible matrices $\mathcal{U,V}$: $\mathcal{A=UBV}$) Does the congruence come from some quiver?
Take $R=k[[x_1,\dots,x_n]]$, where the field $k$ is of zero characteristic (if needed one can take $k=\Bbb{C}$). Let $A$ be a matrix over $R$, consider some "change of coordinates", $\phi\in Aut(R)$. Consider the equivalence $A\rightarrow A Jac_\phi$, where the matrix $Jac_\phi$ is the Jacobian of the coordinate change, its entries are $\partial_j \phi_j$. Can one embed this equivalence into the left-right, as above? (i.e. $A=BJac_\phi$ iff for some associated matrices $\mathcal{A,B}$ and some invertible matrices $\mathcal{U,V}$: $\mathcal{A=UBV}$).
In both cases I'd like to replace the conditions on the transforming matrices by some conditions on the matrices that undergo the transformation. What is the official name for this procedure? For which other equivalences it is known? References?
