So by Theorem 4. 1 we may assume that A is in Hermite normal form [B 01. Since B'[B O [I O is an integral matrix, it follows from our assumption that also B Abstract Hermite Normal Form (HNF) matrices are a standard form of integer matrices used in applications such as lattice based cryptography and integer programming.

Hemite normal form computation, which is useful to solve many set theoretic problems, like like testing membership in a lattice, inclusion or equality between two lattices, computing the union and intersection of lattices, etc. Various types of enumeration algorithms, based on standard algorithmic techniques like the greedy method, or Returns the list [H, U, where H is the Hermite normal form of A, and U is a corresponding transformation matrix.

Return Values Matrix of the same domain type as A. Because the basis is in the columns of the matrix A, the columnstyle Hermite normal form must be used.

Given two bases for a lattice, A and A the equivalence problem is to decide if. This can be done by checking if the columnstyle Hermite normal form of A and A' are the same up to the addition of zero columns. Hermite normal form: ihermite ihermite takes as argument a matrix A with coefficients in. ihermite returns two matrices U and B such that BUA, U is invertible in (det (U) 1) and B is uppertriangular.

Smith normal form, Hermite normal form, polynomial algorithm, Greatest Common Divisor, matrixtriangulation, integer matrices, 1. triangular mite normal form, and therefore, is a lattice. Now, we prove that this basis H is actually unique. We shall need the following simple observation: If H h1, h2,hn is a matrix in Hermite normal form and b 2(H) is a vector with rst i 1 components equal to zero, then h Xn j1 xjhj Xn ji xjhj for some integers xi.

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