Abstract
Special matrices are widely used in information society. The gcd-matrices have be conducted to study over Descartes direct-product of some finite positive integer sets. If Descartes direct-product \( S = S_{1} \times S_{2} \times \cdots \times S_{n} \) with n finite positive integer sets as direct product terms, then S is finite too. Without loss of generality, set \( S = \left\{ {d_{1} ,d_{2} ,\ldots, d_{t} } \right\} \), and \( \forall {\text{a}} = ({\text{a}}_{1} ,{\text{a}}_{2} ,\ldots, {\text{a}}_{n} ),{\text{b}} = ({\text{b}}_{1} ,{\text{b}}_{2} ,\ldots, {\text{b}}_{n} ) \in S \), the general greatest common factor is defined as \( \gcd ({\text{a}},{\text{b}}) = \prod\nolimits_{i = 1}^{n} {\gcd ({\text{a}}_{i} ,{\text{b}}_{i} )} \). And create a square matrix \( \left\langle S \right\rangle = (s_{ij} )_{{{\text{t}} \times {\text{t}}}} = (\gcd (d_{i} ,d_{j} ))_{{{\text{t}} \times {\text{t}}}} \) possessed the general greatest common factors \( \gcd (d_{i} ,d_{j} ) \) as arrays \( s_{ij} = \gcd (d_{i} ,d_{j} ) \). We have researched upper bound and lower bound of the determinant \( \det \left\langle S \right\rangle \) of the \( t \times t \) gcd-matrix \( \left\langle S \right\rangle \), and compute the determinant’s value under special or specific conditions in the article. At last, some well results about the gcd-matrix has been extend from Descartes direct-product of some finite positive integer sets to general direct product of the posets.
Similar content being viewed by others
References
Anderson DD, Izumi S, Ohno Y et al (2016) GCD and LCM-like identities for ideals in commutative rings. J Algebra Appl 15(01):12
Altınışık E, Büyükköse Şerife (2016) On bounds for the smallest and the largest eigenvalues of GCD and LCM matrices. Math Inequal Appl 19(1):117–125
Bourne M, Winkler JR, Su Y (2017) A non-linear structure-preserving matrix method for the computation of the coefficients of an approximate greatest common divisor of two Bernstein polynomials. J Comput Appl Math 320:221–241
Bourne M, Winkler JR, Yi S (2017) The computation of the degree of an approximate greatest common divisor of two Bernstein polynomials. Appl Numer Math 111:17–35
Rajarama Bhat BV (1991) On greatest common divisor matrices and their applications. Linear Algbra Appl 158:77–97
Yun Fan, Hongwei Liu (2002) Group and combination coding. Wuhan University Press, Wuhan
Smith HJS (1875–1876) On the value of a certain arithmetical determinant. Proc Lond Math Soc 7:208–212
Gashkov SB, Sergeev IS (2016) On the additive complexity of GCD and LCM matrices. Math Notes 100(1–2):199–212
Ilmonen Pauliina (2016) On meet hypermatrices and their eigenvalues. Linear Multilinear Algebra 64(5):842–855
Korkee I, Haukkanen P (2001) Bounds for determinants of meet matrices associated with incidence functions. Linear Algebra Appl 329:77–88
Bourque K, Ligh S (1992) On GCD and LCM matrices. Linear Algebra Appl 174:65–74
Bourque K, Ligh S (1993) Matrices associated with arithmetical functions. Linear Multilinear Algebra 34:261–267
Bourque K, Ligh S (1993) Matrices associated with classes of arithmetical function. Number Theory 45:367–376
Leonetti P, Sanna C (2017) On the greatest common divisor of n and the nth Fibonacci number. Roc Mt J Math. Retrieved from https://arxiv.org/pdf/1704.00151
Mctague C (2017) On the Greatest Common Divisor of Binomial Coefficients. The American Mathematical Monthly 124(4):353
Jacobson N (1974) Basic Algebra I. W.H. Freeman, San Francisco
Haukkanen P (1996) On meet matrices on posets. Linear Algebra Appl 249:111–123
Rahman MM, Bhuiyan MNAS, Rahim MS et al (2016) A lightweight PAPR reduction scheme using Greatest Common Divisor matrix based SLM technique. In: IEEE 2016 9th international conference on electrical and computer engineering (ICECE), Dhaka, pp 491–494
Rahman MM, Rahim MS, Bhuiyan MNAS et al (2015) Greatest common divisor matrix based phase sequence for PAPR reduction in OFDM system with low computational overhead. In: International conference on electrical & electronic engineering (ICEEE), Rajshahi, pp 97–100
Stanley RP (1986) Enumerative combinatorics, vol I. Wadsworth and Brooks/cole, Monterey
Sarkar M, Ghosal P (2016) Mathematics using DNA: performing GCD and LCM on a DNA computer. In: IEEE international symposium on nanoelectronic and information systems (iNIS), Gwalior, pp 240–243
Beslin S, Ligh S (1986) Another generalisation of Smith’s determinant. Bull Austral Math Soc 40:413–415
Beslin S, Ligh S (1989) Greatest common divisor matrices. Linear Algebra Appl 118:69–76
Beslin S, Ligh S (1992) GCD-closed set and the determinants of GCD matrices. Fibonacci Quart 30:157–160
Hong S (1998) Bounds for determinants of Matrices associated with classes of arithmetical functions. Linear Algebra Appl 281:311–322
Li Z (1990) The determinants of GCD matrices. Linear Algebra Appl 134:137–143
A lightweight mutual authentication protocol based on elliptic curve cryptography for IoT devices. Int J Adv Intell Paradigms 9(2–3):111–121 (2017)
A lightweight authenticated encryption scheme based on chaotic scml for railway cloud service. IEEE Access 6:711–722 (2017)
ThinORAM: towards practical oblivious data access in fog computing environment. IEEE Trans Serv Comput (2019)
Acknowledgements
This research was supported by the following China projects: 1. Educational research project of Hubei Polytechnic University—Research on the overall optimization to the theory and practice course of “the information theory and Coding”(2015B09); 2. Scientific research project of Hubei Polytechnic University—Quantum coding and its application(16xjz02A), Research on automatic design of cryptographic components(11yjz10R); 3. Scientific research project of Hubei Provincial Department of Education—Automatic design of cryptographic components based on orthographic permutations (B2014041), Quantum error-correcting codes and its application in anti-quantum computation cryptography (D20174502); 4. Scientific research project of Science and Technology Department of Hubei Provincial—Design and application of asymmetric algorithms against quantum computation attack in big data environment (2018CFB550).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
We all declare that we have no conflict of interest in this paper.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Han, H., Li, Q., Wen, Y. et al. Generalization of GCD matrices. Evol. Intel. 15, 2437–2443 (2022). https://doi.org/10.1007/s12065-020-00504-7
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12065-020-00504-7