Hermite normal form

Hermite normal form is a unique form of a rational matrix of full row rank that can be obtained by a series of elementary column operations. It can be computed using row operations and is denoted by the matrix H, which is equal to the product of a unimodular matrix U and the original matrix A. The Hermite normal form theorem states that any rational matrix of full row rank can be brought into this form.From: The Maple Book [2019], Feasible rounding approaches for equality constrained mixed-integer optimization problems [2023], Energy-Aware Memory Management for Embedded Multimedia Systems [2019]

Matrices over Integral Domains

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Purchase Book

Published in Leslie Hogben, Richard Brualdi, Anne Greenbaum, Roy Mathias, Handbook of Linear Algebra, 2006

Shmuel Friedland

A ∈ Fm×n can be brought to its unique Hermite normal form, called the reduced row echelon form (RREF), bini = 1, bjni = 0, j = 1, …, i − 1, i = 1, …, r = rank A,

Feasible rounding approaches for equality constrained mixed-integer optimization problems

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Journal Information

Published in Optimization, 2023

Christoph Neumann, Oliver Stein

Performing any of these operations is equivalent to post-multiplying D by a unimodular matrix (an integer matrix with a determinant of or ). The Hermite normal form theorem states that any rational matrix of full row rank can be brought into a unique HNF by a series of elementary column operations (cf. [7], or, e.g. [8] for a comprehensive introduction). In fact, one may find an unimodular -matrix U such that is the HNF of D. With the algorithm introduced in [9], this may even be done in polynomial time with respect to the length of the binary encoded input data.

Hermite normal form

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Matrices over Integral Domains

Feasible rounding approaches for equality constrained mixed-integer optimization problems