> [!NOTE] Theorem (Every Matrix has an SNF) > Let $A\in\mathbb{R}^{mn}$ be a [[Real Matrices|real matrix]] of order $m\times n.$ Then there exists [[Elementary Matrices|elementary matrices]] $E_{i}\in \text{Mat}_{mm}$ and $F_{i}\in \text{Mat}_{nn}$ so that $E_{k}\dots E_{1}AF_{1}\dots F_{l}$is in [[Smith Normal Form for Real Matrix|Smith normal form]]. **Proof**: By [[Existence of Reduced Row Echelon Form of Real Matrix]], there exists elementary matrices $E_{i}\in\mathbb{R}^{mm}$ so that the matrix $B=E_{k}\dots E_{1}A$ is in RREF. Note the positions of the pivot entries $a_{ij}=1$, for specific $i,j$. Now choose elementary matrices $D_{i}\in\mathbb{F}^{nn}$ so that $D_{l}\dots D_{1}B^{T}$ is in RREF, being careful to use the previous pivot entries $a_{ji}=1$ as the leading $1$s for those columns (this happens automatically if you always choose the top nonzero entry of any nonzero column as the pivot). Then setting $F_{i}=D_{i}^{T}$ gives the result.