Definition#

Smith normal form#

Let \(\mathbf{M}\) be \(m \times n\) integer matrix. There exist some unimodular matrices \(\mathbf{L} \in \mathbb{Z}^{m \times m}\) and \(\mathbf{R} \in \mathbb{Z}^{n \times n}\) such that

\[\begin{split}\mathbf{D} := \mathbf{LMR} = \begin{pmatrix} d_{1} & & \mathbf{O} & \mathbf{0} \\ & \ddots & & \vdots \\ \mathbf{O} & & d_{r} & \mathbf{0} \\ \mathbf{0} & \cdots & \mathbf{0} & \mathbf{O} \end{pmatrix}\end{split}\]

where \(d_{i}\) is positive integer and \(d_{i+1}\) devides \(d_{i}\). Then \(\mathbf{D}\) is called Smith normal form.

Row-style Hermite normal form#

Let \(\mathbf{M}\) be \(m \times n\) integer matrix. It has a row-style Hermite normal form \(\mathbf{H}\) if there exists a unimodular matrices \(\mathbf{L} \in \mathbb{Z}^{m \times m}\) such that \(\mathbf{H}=\mathbf{LM}\) satisfied the following conditions

  1. \(H_{ij} \geq 0 \quad (1 \leq i \leq m, 1 \leq j \leq n)\)

  2. \(H_{ij} = 0 \quad (i > j \, \wedge i > r)\)

  3. \(H_{ij} < H_{jj} \quad (i < j, 1 \leq j \leq r)\)

  4. \(r = \mathrm{rank} \mathbf{A}\)

If \(\mathbf{M}\) is full rank, the Hermite normal form \(\mathbf{H}\) is uniquely determined.

Column-style Hermite normal form#

Let \(\mathbf{M}\) be \(m \times n\) integer matrix. It has a column-style Hermite normal form \(\mathbf{H}\) if there exists a unimodular matrices \(\mathbf{R} \in \mathbb{Z}^{n \times n}\) such that \(\mathbf{H}=\mathbf{MR}\) satisfied the following conditions

  1. \(H_{ij} \geq 0 \quad (1 \leq i \leq m, 1 \leq j \leq n)\)

  2. \(H_{ij} = 0 \quad (i < j \, \wedge j > r)\)

  3. \(H_{ij} < H_{ii} \quad (i > j, 1 \leq i \leq r)\)

  4. \(r = \mathrm{rank} \mathbf{A}\)

If \(\mathbf{M}\) is full rank, the Hermite normal form \(\mathbf{H}\) is uniquely determined.

Reference (Japanese)#

  • 伊理 正夫, 線形代数汎論 (朝倉出版, 2009)