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.