> [!NOTE] Definition (Smith Normal Form) > Let $A$ be a [[Real Matrices|real matrix]]. Then $A$ is in *Smith Normal Form* iff it is both in [[Reduced Row Echelon Form for Real Matrix|RREF]] and [[Reduced Column Echelon Form for Real Matrix|RCEF]]: that is, it has the form: $\begin{align}A&=\begin{pmatrix} I_{r} & 0_{r,n-r} \\ 0_{m-r,r} & 0_{m-r,n-r} \end{pmatrix} \\ &\left.~=~\left(\begin{array}{ccccc|ccc}1&0&0&\cdots&0&0&\cdots&0\\0&1&0&\cdots&0&0&\cdots&0\\\vdots&&&&\vdots&\vdots&&\vdots\\0&0&0&\cdots&1&0&\cdots&0\\\hline0&0&0&\cdots&0&0&\cdots&0\\\vdots&&&&\vdots&\vdots&&\vdots\\0&0&0&\cdots&0&0&\cdots&0\end{array}\right.\right) \end{align}$ > The ***rank*** of this Smith normal form is defined to be the integer $r\geq 0$. > # Properties > [!info] Rank > Note that $r$ is the rank of $A$ since the nonzero columns of the reduced column echelon form of $A$ form a basis of the [[Column Span of Real Matrix|column span]]. > [!NOTE] Theorem (Uniqueness) > The Smith normal of any matrix is unique >See [[Reduced Row Echelon Form for Real Matrix#^f88761|proof]]. # Example We can compute the Smith normal form of the following matrix as follows $\begin{align} &\begin{pmatrix}1&-3&0&1\\0&0&1&2\\-2&0&1&0\end{pmatrix} \stackrel{A^{14}(-1) A^{12}(3) }{ \longrightarrow} \begin{pmatrix}1&0&0&0\\0&0&1&2\\-2&0&1&2\end{pmatrix} \stackrel{S^{24}}{ \longrightarrow} \begin{pmatrix}1&0&0&0\\0&2&1&0\\-2&2&1&0\end{pmatrix} \\ &\stackrel{ A^{23}(-1) M^{2}\left( \frac{1}{2} \right) }{ \longrightarrow} \begin{pmatrix}1&0&0&0\\0&1&0&0\\-2&1&0&0\end{pmatrix} \stackrel{A_{23}(-1) A_{13}(2) }{ \longrightarrow} \begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&0&0\end{pmatrix} \end{align}$ # Applications - [[Rank-nullity formula]].