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In linear algebrathe Hermite normal form is an analogue of reduced echelon form for matrices over the integers Z.

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Other applications of the Hermite normal form include integer programming[1] cryptography[2] and abstract algebra. Various authors may prefer to talk about Hermite normal form in either row-style or column-style. They are essentially the same up to transposition. The third condition is not standard among authors, for example some sources force non-pivots to be nonpositive [7] [8] or place no sign restriction on them.

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A unimodular matrix is a square invertible integer matrix whose determinant is 1 or Note that the row-style definition has a unimodular matrix U multiplying A on the left meaning U is acting on the rows of Awhile the column-style definition has the unimodular matrix action on the columns of A. The two definitions of Hermite normal forms are simply transposes of each other. There are many algorithms for computing the Hermite normal form dating back to It was not until when an algorithm for computing the Hermite normal form that ran in strongly polynomial time was first developed; [13] that is, the number of steps to compute the Hermite normal form is bounded above by a polynomial in the dimensions of the input matrix, and the space used by the algorithm intermediate numbers Normal form berechnen online dating bounded Normal form berechnen online dating a polynomial in the binary encoding size of the numbers in the input matrix.

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One class of algorithms is based on Gaussian elimination in that special elementary matrices are repeatedly used. If the columns of a matrix A are the a ithe lattice can be associated with the columns of a matrix, and A is said to be a basis of L.

Because the Hermite normal form is unique, it can be used to answer many questions about two lattice descriptions. Because the basis is in the columns of the matrix Athe column-style Hermite normal form must be used.

From Wikipedia, the free encyclopedia. Linear Algebra and its Applications.