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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| Format: | Article |
| Language: | English |
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MDPI AG
2021
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| Online Access: | http://psasir.upm.edu.my/id/eprint/95139/ http://psasir.upm.edu.my/id/eprint/95139/1/Square%20integer%20matrix%20with%20a%20single%20non-integer%20entry%20in%20its%20inverse.pdf |
| _version_ | 1848862080833159168 |
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| author | Mandangan, Arif Kamarulhaili, Hailiza Asbullah, Muhammad Asyraf |
| author_facet | Mandangan, Arif Kamarulhaili, Hailiza Asbullah, Muhammad Asyraf |
| author_sort | Mandangan, Arif |
| building | UPM Institutional Repository |
| collection | Online Access |
| 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. |
| first_indexed | 2025-11-15T13:11:21Z |
| format | Article |
| id | upm-95139 |
| institution | Universiti Putra Malaysia |
| institution_category | Local University |
| language | English |
| last_indexed | 2025-11-15T13:11:21Z |
| publishDate | 2021 |
| publisher | MDPI AG |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | upm-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 |
| spellingShingle | Mandangan, Arif Kamarulhaili, Hailiza Asbullah, Muhammad Asyraf Square integer matrix with a single non-integer entry in its inverse |
| title | 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_short | 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 |
| url | http://psasir.upm.edu.my/id/eprint/95139/ http://psasir.upm.edu.my/id/eprint/95139/ http://psasir.upm.edu.my/id/eprint/95139/ http://psasir.upm.edu.my/id/eprint/95139/1/Square%20integer%20matrix%20with%20a%20single%20non-integer%20entry%20in%20its%20inverse.pdf |