On the Hadamard product of GCD and LCM matrices associated with semi-multiplicative functions

Ayse Nalli and Pentti Haukkanen
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 9, 2003, Number 4, Pages 90–98
Full paper (PDF, 77 Kb)

Details

Authors and affiliations

Ayse Nalli
Department of Mathematics, Faculty of Science and Literature, University of Selcuk
42070 Campüs Konya, Turkey

Pentti Haukkanen
Department of Mathematics, Statistics and Philosophy,
FIN-33014 University of Tampere, Finland

Abstract

Let S = {x₁, x₂, …, x_n} be an ordered set of distinct positive integers with x₁ < x₂ < … < x_n (n > 1). We provide some properties for the n × n matrix (G)_f o [L]_f on S, where (G)_f denotes the n × n GCD matrix on S associated with f, [L]_f denotes the n × n LCM matrix on S associated with f and o denotes the Hadamard product.

Keywords

• Hadamard product
• GCD matrix
• LCM matrix
• Semi-multiplicative function

AMS Classification

• 11C20
• 11A05
• 15A36
• 11A25

References

1. S. J. Beslin, Reciprocal GCD matrices and LCM matrices.  Fibonacci Quart. 29 (1991), no. 3, 271-274.
2. P. Haukkanen  and  J. Sillanpää, Some analogues of Smith’s  determinant, Linear  and Multilinear Algebra 41 (1996),  233-244.
3. P. Haukkanen, J. Wang and J. Sillanpää,  On Smith’s determinant.  Linear Algebra Appl.  258  (1997), 251-269.
4. R. A. Horn and C. A. Johnson, Matrix Analysis, Cambridge University Press, New York 1985.
5. I. Korkee and P. Haukkanen, On meet and join matrices associated with incidence functions. Linear Algebra Appl.  372C (2003), 127-153.
6. R. Sivaramakrishnan, Classical Theory of Arithmetic Functions, Marcel Dekker, New York, 1989.
7. H. J. S. Smith, On the value of a certain arithmetical determinant. Proc. London Math. Soc. 7 (1875/76), 208-212.
8. N. Tuglu and D. Tasci, On the LCM-reciprocal GCD matrices. Far East J. Math. Sci. (FJMS) 6 (2002), no. 1, 89-94.