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Hermite normal form

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Context

Linear algebra

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Contents

Definition

Definition

(Hermite normal form)

A matrix

HMat n×m() H \in Mat_{n \times m}(\mathbb{Z})

with integer entries (nn rows, mm columns) is said to be in Hermite normal form if the following conditions hold:

  1. All entries are non-negative numbers \in \mathbb{N} \subset \mathbb{Z}:

    H ij0. H_{i j} \geq 0 \,.
  2. The first non-vanishing entry p iH i,j ip_{i} \coloneqq H_{i,j_{i}} in each row is strictly to the right of the first non-vanishing entry p i1=H i,j i1p_{i-1} = H_{i,j_{i-1}} of the previous row (if any):

    j i>j i1 j_{ i} \gt j_{i - 1}

    These entries are called the pivots of HH.

  3. Each pivot p ip_i is strictly larger than all entries above it in the same row:

    For 1i<i1 \leq i' \lt i we have H i,j i<p i=H i,j iH_{i', j_i} \lt p_i = H_{i, j_i}.

(e.g. Mader 00, theorem 1.5.1)

Remark

There are slightly different conventions in use for Def. , e.g. some authors require the non-vanishing non-pivotal entries to be negative instead of positive. These conventions are equivalent, in that the uniqueness of Hermite normal form, Theorem below applies with respect to each of them.

Properties

Uniqueness

Theorem

(uniqueness of Hermite normal form)

For

MMat n×m() M \in Mat_{n \times m}(\mathbb{Z})

a matrix with integer entries, there exists a unique invertible matrix with integer coefficients

UGL(n,)Mat n×n() U \;\in\; GL(n, \mathbb{Z}) \subset Mat_{n \times n}(\mathbb{Z})

such that the matrix product

HUM H \;\coloneqq\; U \cdot M

has Hermite normal form (Def. ).

Existence follows by integer row reduction of integer matrices, see e.g. Gilbert-Pathria 90, p. 3/4. Uniqueness requires more work (see e.g. Mader 00, theorem 1.5.2).

References

Integer row reduction for integer matrices via greatest common divisors is explained for instance in

  • William Gilbert, Anu Pathria, Linear Diophantine Equations, Dept. Pure Math. Univ. Waterloo, Ontario, Canada, 1990 (pdf, pdf)

Discussion of the general theory of integer matrices includes

  • A. Mader, section 1.5 of Almost completely decomposable groups, CRC Press 2000

Computational implementation is discussed in

  • George Labahn, Vincent Neiger, Wei Zhou, Fast, deterministic computation of the Hermite normal form and determinant of a polynomial matrix (arXiv:1607.04176)

See also

Last revised on October 4, 2018 at 23:23:58. See the history of this page for a list of all contributions to it.