Number of Matrix Equivalence Classes

Theorem
Let $$K$$ be a field.

Let $$\mathcal {M}_{K} \left({m, n}\right)$$ be the $m \times n$ matrix space over $$K$$.

Let $$\mathbf{A}$$ be an $m \times n$ matrix of rank $$r$$ over $$K$$.

Then:

$$\mathbf{A} \equiv \begin{cases} \left[{0_K}\right]_{m n} & : r = 0 \\ & \\ \begin{bmatrix} \mathbf{I}_r & \mathbf{0} \\ \mathbf{0} & \mathbf{0} \end{bmatrix} & : 0 < r < \min \left\{{n, m}\right\} \\ & \\ \begin{bmatrix} \mathbf{I}_r & \mathbf{0} \end{bmatrix} & : r = m < n \\ & \\ \begin{bmatrix} \mathbf{I}_r \\ \mathbf{0} \end{bmatrix} & : r = n < m \\ & \\ \mathbf{I}_r & : r = m = n \end{cases} $$

Thus there are exactly $$\min \left\{{m, n}\right\} + 1$$ equivalence classes for the relation of equivalence on $$\mathcal {M}_{K} \left({m, n}\right)$$, one of which contains only the zero matrix.

Proof
Follows from Equivalent Matrices have Equal Rank.