Square integer matrix with a single non-integer entry in its inverse

Arif Mandangan and Hailiza Kamarulhaili and Muhammad Asyraf Asbullah (2021) Square integer matrix with a single non-integer entry in its inverse. Mathematics, 9. pp. 1-11. ISSN 22277390

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Matrix inversion is one of the most significant operations on a matrix. For any non-singular matrix A?Z^{nÃn}, the inverse of this matrix may contain countless numbers of non-integer entries. These entries could be endless floating-point numbers. Storing, transmitting, or operating such an inverse could be cumbersome, especially when the size n is large. The only square integer matrix that is guaranteed to have an integer matrix as its inverse is a unimodular matrix U?Z^{nÃn}. With the property that det(U)=±1, then U^{-1}?Z^{nÃn} is guaranteed such that UU^{-1}=I, where I?Z^{nÃn} is an identity matrix. In this paper, we propose a new integer matrix \tilde{G}?Z^{nÃn}, which is referred to as an almost-unimodular matrix. With det(\tilde{G})?±1, the inverse of this matrix, \tilde{G}^{-1}?R^{nÃn}, is proven to consist of only a single non-integer entry. The almost-unimodular matrix could be useful in various areas, such as lattice-based cryptography, computer graphics, lattice-based computational problems, or any area where the inversion of a large integer matrix is necessary, especially when the determinant of the matrix is required not to equal ±1. Therefore, the almost-unimodular matrix could be an alternative to the unimodular matrix.

Item Type: Article
Keyword: Square integer matrix , Inversion of integer matrix , Unimodular matrix , Algebraic number theory , Lattice-based cryptography.
Subjects: Q Science > QA Mathematics > QA1-939 Mathematics > QA440-699 Geometry. Trigonometry. Topology
T Technology > T Technology (General) > T1-995 Technology (General) > T55.4-60.8 Industrial engineering. Management engineering > T57-57.97 Applied mathematics. Quantitative methods
Department: FACULTY > Faculty of Science and Natural Resources
Date Deposited: 23 Aug 2023 15:29
Last Modified: 23 Aug 2023 15:29
URI: https://eprints.ums.edu.my/id/eprint/36114

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