Square Integer Matrix with a Single Non-Integer Entry in Its Inverse

Arif Mandangan, Hailiza Kamarulhaili and Muhammad Asyraf Asbullah
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Arif Mandangan: Mathematics, Real-Time Graphics and Visualization Research Laboratory, Faculty of Science and Natural Resources, Universiti Malaysia Sabah, Jalan UMS, Kota Kinabalu 88400, Malaysia
Hailiza Kamarulhaili: School of Mathematical Sciences, Universiti Sains Malaysia, Penang 11800, Malaysia
Muhammad Asyraf Asbullah: Laboratory of Cryptography, Analysis and Structure, Institute for Mathematical Research, Universiti Putra Malaysia, Serdang 43400, Malaysia

Mathematics, 2021, vol. 9, issue 18, 1-12

Abstract: Matrix inversion is one of the most significant operations on a matrix. For any non-singular matrix A ? Z n × 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 ? Z n × n . With the property that det ( U ) = ± 1 , then U ? 1 ? Z n × n is guaranteed such that U U ? 1 = I , where I ? Z n × n is an identity matrix. In this paper, we propose a new integer matrix G ˜ ? Z n × n , which is referred to as an almost-unimodular matrix. With det ( G ˜ ) ? ± 1 , the inverse of this matrix, G ˜ ? 1 ? R n × 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.

Keywords: square integer matrix; inversion of integer matrix; unimodular matrix; algebraic number theory; lattice-based cryptography (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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