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

Matrix inversion is one of the most significant operations on a matrix. For any non-singular matrix A∈Zn×n, 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 coul...

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Main Authors: Mandangan, Arif, Kamarulhaili, Hailiza, Asbullah, Muhammad Asyraf
Format: Article
Language:English
Published: MDPI AG 2021
Online Access:http://psasir.upm.edu.my/id/eprint/95139/1/Square%20integer%20matrix%20with%20a%20single%20non-integer%20entry%20in%20its%20inverse.pdf
http://psasir.upm.edu.my/id/eprint/95139/
https://www.mdpi.com/2227-7390/9/18/2226
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spelling my.upm.eprints.951392023-01-06T08:26:45Z http://psasir.upm.edu.my/id/eprint/95139/ Square integer matrix with a single non-integer entry in its inverse Mandangan, Arif Kamarulhaili, Hailiza Asbullah, Muhammad Asyraf Matrix inversion is one of the most significant operations on a matrix. For any non-singular matrix A∈Zn×n, 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∈Zn×n. With the property that det(U)=±1, then U−1∈Zn×n is guaranteed such that UU−1=I, where I∈Zn×n is an identity matrix. In this paper, we propose a new integer matrix G˜∈Zn×n, which is referred to as an almost-unimodular matrix. With det(G˜)≠±1, the inverse of this matrix, G˜−1∈Rn×n, 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. MDPI AG 2021-09-10 Article PeerReviewed text en http://psasir.upm.edu.my/id/eprint/95139/1/Square%20integer%20matrix%20with%20a%20single%20non-integer%20entry%20in%20its%20inverse.pdf Mandangan, Arif and Kamarulhaili, Hailiza and Asbullah, Muhammad Asyraf (2021) Square integer matrix with a single non-integer entry in its inverse. Mathematics, 9 (18). art. no. 2226. 01-11. ISSN 2227-7390 https://www.mdpi.com/2227-7390/9/18/2226 10.3390/math9182226
institution Universiti Putra Malaysia
building UPM Library
collection Institutional Repository
continent Asia
country Malaysia
content_provider Universiti Putra Malaysia
content_source UPM Institutional Repository
url_provider http://psasir.upm.edu.my/
language English
description Matrix inversion is one of the most significant operations on a matrix. For any non-singular matrix A∈Zn×n, 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∈Zn×n. With the property that det(U)=±1, then U−1∈Zn×n is guaranteed such that UU−1=I, where I∈Zn×n is an identity matrix. In this paper, we propose a new integer matrix G˜∈Zn×n, which is referred to as an almost-unimodular matrix. With det(G˜)≠±1, the inverse of this matrix, G˜−1∈Rn×n, 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.
format Article
author Mandangan, Arif
Kamarulhaili, Hailiza
Asbullah, Muhammad Asyraf
spellingShingle Mandangan, Arif
Kamarulhaili, Hailiza
Asbullah, Muhammad Asyraf
Square integer matrix with a single non-integer entry in its inverse
author_facet Mandangan, Arif
Kamarulhaili, Hailiza
Asbullah, Muhammad Asyraf
author_sort Mandangan, Arif
title Square integer matrix with a single non-integer entry in its inverse
title_short Square integer matrix with a single non-integer entry in its inverse
title_full Square integer matrix with a single non-integer entry in its inverse
title_fullStr Square integer matrix with a single non-integer entry in its inverse
title_full_unstemmed Square integer matrix with a single non-integer entry in its inverse
title_sort square integer matrix with a single non-integer entry in its inverse
publisher MDPI AG
publishDate 2021
url http://psasir.upm.edu.my/id/eprint/95139/1/Square%20integer%20matrix%20with%20a%20single%20non-integer%20entry%20in%20its%20inverse.pdf
http://psasir.upm.edu.my/id/eprint/95139/
https://www.mdpi.com/2227-7390/9/18/2226
_version_ 1754531209443016704
score 13.211869

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