Definition
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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
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
\(H_{ij} \geq 0 \quad (1 \leq i \leq m, 1 \leq j \leq n)\)
\(H_{ij} = 0 \quad (i > j \, \wedge i > r)\)
\(H_{ij} < H_{jj} \quad (i < j, 1 \leq j \leq r)\)
\(r = \mathrm{rank} \mathbf{A}\)
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
\(H_{ij} \geq 0 \quad (1 \leq i \leq m, 1 \leq j \leq n)\)
\(H_{ij} = 0 \quad (i < j \, \wedge j > r)\)
\(H_{ij} < H_{ii} \quad (i > j, 1 \leq i \leq r)\)
\(r = \mathrm{rank} \mathbf{A}\)
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)