I'm trying to characterize equivalence classes of matrices over a vector space.

Specifically, let $V$ be a vector space over a field $K$, let $M \in M_n(V)$ be an $n \times n$ matrix with entries in $V$ and let $A,B \in M_n(K)$ be $n \times n$ matrices with entries in $K$. I want to characterize the equivalence class formed by all matrices formed as

$$A M B$$

i.e, equivalent matrices, but over a vector field instead of a ring, and the invertible "side" matrices are picked from the base field, not the vector field itself.

The problem is related to constructing the Jacobian variety of a curve.

So far, I've been experimenting with the $2\times2$ case:

$$\left[\matrix{v_1 & v_3 \\ v_2 & v_4}\right]$$

Let's multiply the equation out:

$$AMB = \left[\matrix{a & c \\ b & d}\right] \left[\matrix{v_1 & v_3 \\ v_2 & v_4}\right] \left[\matrix{e & g \\ f & h}\right]$$

$$=\left[\matrix{eav_1 + ebv_2 + gav_3 + gbv_4 && fav_1+fbv_2+hav_3+hbv_4 \\ ecv_1+edv_2 +gcv_3+gdv_4 && fcv_1+fdv_2+hcv_3+hdv_4}\right]$$

Obviously, all of the vectors have to be in the linear space spanned by the original four vectors. So, is picking a four-dimensional subspace of $V$ enough to specify our equivalence class?

No. Let the transformed matrix have the form:

$$\left[\matrix{ A_1v_1 + A_2v_2 + A_3v_3 + A_4v_4 && C_1v_1 + C_2v_2 + C_3v_3 + C_4v_4 \\ B_1v_1 + B_2v_2 + B_3v_3 + B_4v_4 && D_1v_1 + D_2v_2 + D_3v_3 + D_4v_4 }\right]$$

We see that the obvious equality $ea\cdot gb = eb \cdot ga$ implies that $A_1A_4 = A_2A_3$. Likewise, $B_1B_4=B_2B_3$, $C_1C_4=C_2C_3$ and $D_1D_4=D_2D_3$.

There are also similar cross-relationships between the four vectors, but the restriction I just derived is enough to show that a four-dimensional subspace alone isn't enough to specify an equivalence class. The vectors have to picked from a particular three-dimensional subvariety, and it's a subvariety, not a subspace, because the relationships $A_1A_4 = A_2A_3$, etc., are not linear.

So, specifying a equivalence class requires specifying a four-dimensional subspace, then picking a certain three-dimensional subvariety from that space. I've left out considerations like the vectors being linearly dependent, but I think this is enough to give the sense of what I'm trying to accomplish.

Has anybody seen anything like this? Any idea how to characterize these equivalence classes?