Vol. 4, No. 1, 2020

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A canonical form for positive definite matrices

Mathieu Dutour Sikirić, Anna Haensch, John Voight and Wessel P.J. van Woerden

Vol. 4 (2020), No. 1, 179–195
Abstract

We exhibit an explicit, deterministic algorithm for finding a canonical form for a positive definite matrix under unimodular integral transformations. We use characteristic sets of short vectors and partition-backtracking graph software. The algorithm runs in a number of arithmetic operations that is exponential in the dimension n, but it is practical and more efficient than canonical forms based on Minkowski reduction.

Keywords
canonical form, quadratic form, positive definite matrix, lattice isomorphism, graph isomorphism
Mathematical Subject Classification 2010
Primary: 11H55, 11H56, 15A21
Milestones
Received: 28 February 2020
Accepted: 29 April 2020
Published: 29 December 2020
Authors
Mathieu Dutour Sikirić
Institut Rudjer Bošković
Zagreb
Croatia
Anna Haensch
Department of Mathematics and Computer Science
Duquesne University
Pittsburgh, PA
United States
John Voight
Department of Mathematics
Dartmouth College
Hanover, NH
United States
Wessel P.J. van Woerden
Centrum Wiskunde & Informatica (CWI)
Amsterdam
Netherlands