Similarity classes of matrices Let $M_n(K)$ be the set of all $n\times n$ matrices over a field $K$. If $\mathcal{R}$ is the equivalence relation defined by matrix similarity, what does the quotient $M_n(K)/\mathcal{R}$ looks like? Is there something that characterizes it in terms of cardinality? Is there a way to extend matrix operations to equivalence classes, making it an algebraic quotient structure of $M_n(K)$? Is there a good interpretation of $M_n(K)/\mathcal{R}$ in terms of similarity invariants (e.g., matrix determinants)?
Today, on my linear algebra class, we were introduced the concept of determinants. Our teacher used geometric isometry invariants (such as area and volume) to introduce the motivation for matrix determinants. I then asked myself what would be the generalized interpretation of the determinant as some type of ''invariant''. Some Google search led me to matrix similarity, and my algebraic intuition says that this information would be codified in the quotient structure - once we ''kill' the big structure by the invariants, we would find exactly what is not varying.
Well, though I never though about verifying it for any pathological case, I believe that the map $\det: M_n(K) \to K$ is surjective. That would mean that the quotient $M_n(K)/\mathcal{R}$ is at least the same cardinality as $K$ (because similar matrices have the same determinant, but I'm not sure about the converse. If it is true, matrix with the same determinant are similar, then cardinality equality would follow).
I'm not sure how the operations can be extended. Maybe the quotient can be seen as a vector space over $K$ as well: scalar multiplication is surely well defined, but I'm not sure about the sum. If I restrict myself to some subset of $M_n(K)$, like the general linear group or the special linear group, could I get something more?
And the interpretation or mathematical application is exactly what I long for.
 A: Let $\mathcal C$ be the set of $n\times n$ matrices over $K$ which are in rational canonical form. Since every element of $M_n(K)$ is similar to a unique matrix in rational canonical form, you can identify $M_n(K)/\mathcal R$ with $\mathcal C$. In particular, the cardinality of $M_n(K)/\mathcal R$ is the number of $n\times n$ matrices over $K$ in rational canonical form, i.e. the number of ways to construct a matrix
$$
\begin{bmatrix}
0&0&\dots&\dots&\dots&-b_0\\
1 &0&\dots&\dots&\dots&-b_1\\
0&1&\dots&\dots&\dots&-b_2\\
0&0&\ddots& &&\vdots\\
\vdots&\vdots&&\ddots&&\vdots\\
0&0&\dots&\dots&1&-b_{n-1}
\end{bmatrix}.
$$
There are $n$ choices for $b_0,\dots,b_{n-1}$, and so
$$
|M_n(K)/\mathcal R|=|K^n|.
$$
As for your reasoning about the determinant: it is true that $\det:M_n(K)\to K$ is a surjective homomorphism. Given $x\in K$, the matrix
$$
\begin{bmatrix}
x&0&\dots&0\\
0&1&\dots&0\\
\vdots&\vdots&\ddots&\vdots\\
0&0&\dots&1
\end{bmatrix}
$$
has determinant $x$. Since similar matrices have the same determinant, you are right that $|M_n(K)/\mathcal R|\geq |K|$. However, the converse is false: matrices with the same determinant might not be similar. For example,
$$
SO(2)=\left\{\begin{bmatrix}\cos\theta &-\sin\theta\\\sin\theta&\cos\theta\end{bmatrix}:\theta\in [0,2\pi)\right\}
$$
consists of rotation matrices with determinant $1$, but no two distinct matrices in $SO(2)$ are similar (one way to see prove this is to note that the eigenvalues of a rotation matrix by $\theta$ are $e^{\pm i\theta}$).
