Abstract

We identify and analyse obstructions to factorisation of integer matrices into products NTN or N2 of matrices with rational or integer entries. The obstructions arise as quadratic forms with integer coefficients and raise the question of the discrete range of such forms. They are obtained by considering matrix decompositions over a superalgebra. We further obtain a formula for the determinant of a square matrix in terms of adjugates of these matrix decompositions, as well as identifying a co-Latin symmetry space.

Highlights

  • IntroductionWe establish in Corollary 3.1 that the existence of integer solutions to a certain quadratic equation is a necessary condition for a matrix factorisation of the type M = N 2 or M = N T N (for symmetric positive definite M ) to exist

  • The question whether a given square integer matrix M can be factorised into a product of two integer matrices, either in the form of a square M = N 2 or in the form M = N T N, has a long history in number theory

  • The number theoretic properties relating to the factorisation of symmetric positive definite n × n integer matrices M with fixed determinant have classical connections to the theory of positive definite quadratic forms in n variables, see e.g. [12] and the above references

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Summary

Introduction

We establish in Corollary 3.1 that the existence of integer solutions to a certain quadratic equation is a necessary condition for a matrix factorisation of the type M = N 2 or M = N T N (for symmetric positive definite M ) to exist.

Integer factorisation of matrices and the quadratic form obstruction
Co-Latin matrices
Conclusions

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