#### Vol. 4, No. 1, 2020

 Recent Volumes 4: ANTS XIV 3: Hillman: Poincaré Duality 2: ANTS XIII 1: ANTS X
 The Open Book Series All Volumes About the Series Ethics Statement Purchase Printed Copies Author Index MSP Books and Monographs Other MSP Publications
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