> [!NOTE] Lemma > Let $A$ be a [[Real Matrices|real matrix]] of order $m\times n.$ Let $EAF$ denote its [[Smith Normal Form for Real Matrix|Smith normal form]]. Then it is unique. **Proof**: The rank of the smith normal form $EAF$ of $A$ is equal to the number of nonzero columns, which is the same as $\dim \text{Im }L_{EAF},$ which equals $\dim \text{Im }L_{A}$ and so is independent of which elementary products of $E$ and $F$ you choose.