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.