A matrix of rank k is equivalent to a matrix with 1 in the first k diagonal entries and 0 elsewhere.

Theorem: Every $m \times n$ matrix $A$ of rank $k$ is equivalent to the canonical form:

$$D_k = \begin{pmatrix} I_k & 0 \\ 0 & 0 \end{pmatrix}$$

where $I_k$ is the $k \times k$ identity matrix and all other entries are zero.

Proof sketch: By row and column operations (which correspond to left- and right-multiplication by invertible matrices), any matrix can be reduced to this form. This is essentially the result of Gauss-Jordan elimination extended to column operations.

Example: A $3 \times 4$ matrix of rank 2 is equivalent to:
$$\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix}$$

This canonical form shows that rank is the only invariant under matrix equivalence: two matrices are equivalent if and only if they have the same rank.