Integer matrix factorisations, superalgebras and the quadratic form obstruction

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

doi:10.1016/j.laa.2021.03.028

Cited by 5 publications

(8 citation statements)

References 10 publications

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“…Additionally, as established by the authors in conjunction with Schmidt [16], if M is symmetric, then in this decomposition the odd part M V takes the form M V = a1 T n + 1 n a T , for some vector a ∈ R n , with 1 n the n-dimensional vector with every entry 1. The vector a can be obtained explicitly from the formula…”

Section: Methodsmentioning

confidence: 91%

“…For the Wilson matrix the weight w W = 119 16 . Another important concept is the idea of the matrix M having a unique decomposition M = M V + M S + w M E n over the type V and type S matrix symmetry spaces [16], where E n is the n × n matrix with every entry 1. We say that M V = (m ij ) has the (type V) vertex cross-sum property if for i = i ′ and j = j ′ , we have…”

Section: Methodsmentioning

confidence: 99%

“…In particular, if M is an integer matrix, then the vector n 2 a has integer entries. As established in [16], a necessary (but not sufficient) condition for an integer matrix M to be factored as M = Z T Z is that the weights in the S ⊕ V decompositions of the matrix and the matrix factor equate. Setting Z = Z V + Z S + w Z E n with Z V = a1 T n + 1 n a T , we obtain the quadratic equation [16, Theorem 3.1]…”

Section: Methodsmentioning

confidence: 99%

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Optimizing and Factorizing the Wilson Matrix

Higham¹,

Lettington²

2022

The American Mathematical Monthly

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The Wilson matrix, W , is a 4 × 4 unimodular symmetric positive definite matrix of integers that has been used as a test matrix since the 1940s, owing to its mild ill-conditioning. We ask how close W is to being the most ill-conditioned matrix in its class, with or without the requirement of positive definiteness. By exploiting the matrix adjugate and applying various matrix norm bounds from the literature we derive bounds on the condition numbers for the two cases and we compare them with the optimal condition numbers found by exhaustive search. We also investigate the existence of factorizations W = Z T Z with Z having integer or rational entries. Drawing on recent research that links the existence of these factorizations to number-theoretic considerations of quadratic forms, we show that W has an integer factor Z and two rational factors, up to signed permutations. This little 4 × 4 matrix continues to be a useful example on which to apply existing matrix theory as well as being capable of raising challenging questions that lead to new results.

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Section: Methodsmentioning

confidence: 91%

Section: Methodsmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

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Optimizing and Factorizing the Wilson Matrix

Higham¹,

Lettington²

2022

The American Mathematical Monthly

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“…Finding an integer factor Z is a nontrivial task. Higham, Lettington, and Schmidt [33] have recently derived conditions for integer factorizations to exist and have developed an approach to computing them. A case in point is the Wilson matrix, a moderately ill-conditioned symmetric positive definite matrix that has a long history as a test matrix.…”

Section: Specific Matricesmentioning

confidence: 99%

“…A case in point is the Wilson matrix, a moderately ill-conditioned symmetric positive definite matrix that has a long history as a test matrix. An integer factor was discovered in [32] and two ratio-nal factors were discovered in [33]. The matrix and its factors are available through core/wilson:…”

Section: Specific Matricesmentioning

confidence: 99%

Anymatrix: an extensible MATLAB matrix collection

Higham

Mikaitis

2021

Numer Algor

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Anymatrix is a MATLAB toolbox that provides an extensible collection of matrices with the ability to search the collection by matrix properties. Each matrix is implemented as a MATLAB function and the matrices are arranged in groups. Compared with previous collections, Anymatrix offers three novel features. First, it allows a user to share a collection of matrices by putting them in a group, annotating them with properties, and placing the group on a public repository, for example on GitHub; the group can then be incorporated into another user’s local Anymatrix installation. Second, it provides a tool to search for matrices by their properties, with Boolean expressions supported. Third, it provides organization into sets, which are subsets of matrices from the whole collection appended with notes, which facilitate reproducible experiments. Anymatrix comes with 146 built-in matrices organized into 7 groups with 49 recognized properties. The authors continue to extend the collection and welcome contributions from the community.

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Symmetric bilinear forms, superalgebras and integer matrix factorization

Fretwell,

Roberts

2024

Linear Algebra and its Applications

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Integer matrix factorisations, superalgebras and the quadratic form obstruction

Cited by 5 publications

References 10 publications

Optimizing and Factorizing the Wilson Matrix

Optimizing and Factorizing the Wilson Matrix

Anymatrix: an extensible MATLAB matrix collection

Symmetric bilinear forms, superalgebras and integer matrix factorization

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