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

Put content here**Theorem:** Every $m \times n$ matrix $A$ of rank $k$ is equivalent to the canonical form:
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$$D_k = \begin{pmatrix} I_k & 0 \\ 0 & 0 \end{pmatrix}$$
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where $I_k$ is the $k \times k$ identity matrix and all other entries are zero.
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**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.
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**Example:** A $3 \times 4$ matrix of rank 2 is equivalent to:
$$\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix}$$
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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.

# Parents

* Matrix equivalence