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Integer matrix factorisations, superalgebras and the quadratic form obstruction

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Nicholas J. Higham, Matthew C. Lettington, Karl Michael Schmidt

Number Theory · Submitted 6 Mar 2021

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We identify and analyse obstructions to factorisation of integer matrices into products $N^T N$ or $N^2$ of matrices with rational or integer entries.

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arXiv ID2103.04149 (version 1)
Pith IDPARETZJ4, Pith's permanent record of this paper and its review
CitationKarl Michael Schmidt, Matthew C. Lettington, Nicholas J. Higham. "Integer matrix factorisations, superalgebras and the quadratic form obstruction." arXiv:2103.04149 (2021). https://pith.science/paper/PARETZJ4
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,595 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.22 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.10 · compares with prior results 0.04 · mentions code or data availability 0.02.

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arXiv listing abstract v1. Results and the main claim are underlined; .

We identify and analyse obstructions to factorisation of integer matrices into products $N^T N$ or $N^2$ 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 $\it co-Latin$ symmetry space.

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 We identify and analyse obstructions to factorisation of integer matrices into products $N^T N$ or $N^2$ of matrices with rational or integer entries. contribution 1.00 0.95 0.45 asserted 1.00
s1 The obstructions arise as quadratic forms with integer coefficients and raise the question of the discrete range of such forms. result 0.53 0.69 0.15 asserted 1.00
s2 They are obtained by considering matrix decompositions over a superalgebra. method 0.99 0.84 0.07 asserted 1.00
s3 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 $\it co-Latin$ symmetry space. result 0.76 0.94 0.17 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

We identify and analyse obstructions to factorisation of integer matrices into products $N^T N$ or $N^2$ 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 $\it co-Latin$ symmetry space.

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