Dot Products and Matrix Properties of 4×4 Strongly Magic Squares
International Journal of Theoretical and Applied Mathematics
Volume 3, Issue 2, April 2017, Pages: 64-69
Received: Nov. 4, 2016; Accepted: Dec. 27, 2016; Published: Feb. 13, 2017
Views 3392      Downloads 117
Authors
Neeradha. C. K., Dept. of Science & Humanities, Mar Baselios College of Engineering & Technology, Thiruvananthapuram, Kerala, India
V. Madhukar Mallayya, Department of Mathematics, Mohandas College of Engineering & Technology, Thiruvananthapuram, Kerala, India
Article Tools
Follow on us
Abstract
Magic squares have been known in India from very early times. The renowned mathematician Ramanujan had immense contributions in the field of Magic Squares. A magic square is a square array of numbers where the rows, columns, diagonals and co-diagonals add up to the same number. The paper discuss about a well-known class of magic squares; the strongly magic square. The strongly magic square is a magic square with a stronger property that the sum of the entries of the sub-squares taken without any gaps between the rows or columns is also the magic constant. In this paper a generic definition for Strongly Magic Squares is given. The matrix properties of 4×4 strongly magic squares dot products and different properties of eigen values and eigen vectors are discussed in detail.
Keywords
Strongly Magic Square (SMS), Dot Products of SMS, Eigen Values of SMS, Rank and Determinant of SMS
To cite this article
Neeradha. C. K., V. Madhukar Mallayya, Dot Products and Matrix Properties of 4×4 Strongly Magic Squares, International Journal of Theoretical and Applied Mathematics. Vol. 3, No. 2, 2017, pp. 64-69. doi: 10.11648/j.ijtam.20170302.13
Copyright
Copyright © 2017 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
References
[1]
Schuyler Cammann, Old Chinese magic squares. Sinologica 7 (1962), 14–53.
[2]
Andrews, W. S. Magic Squares and Cubes, 2nd rev. ed. New York: Dover, 1960.
[3]
Claudia Zaslavsky, Africa Counts: Number and Pattern in African Culture. Prindle, Weber & Schmidt, Boston, 1973.
[4]
Paul C. Pasles. Benjamin Franklin’s numbers: an unsung mathematical odyssey. Princeton UniversityPress, Princeton, N. J., 2008.
[5]
C. Pickover. The Zen of Magic Squares, Circles and Stars. Princeton University Press, Princeton, NJ, 2002.
[6]
Bruce C. Berndt, Ramanujan’s Notebooks Part I, Chapter 1 (pp 16-24), Springer, 1985.
[7]
T. V. Padmakumar “Strongly Magic Square”, Applications Of Fibonacci Numbers Volume 6 Proceedings of The Sixth International Research Conference on Fibonacci Numbers and Their Applications, April 1995.
[8]
Charles Small, “Magic Squares Over Fields” The American Mathematical Monthly Vol. 95, No. 7 (Aug. - Sep., 1988), pp. 621-625.
[9]
Neeradha. C. K, Dr. V. Madhukar Mallayya “Generalized Form Of A 4x4 Strongly Magic Square” IJMMS, Vol. 12, No. 1 (January-June; 2016), pp 79-84.
[10]
A. Mudgal, Counting Magic Squares, Undergraduate thesis, IIT Bombay, 2002.
ADDRESS
Science Publishing Group
1 Rockefeller Plaza,
10th and 11th Floors,
New York, NY 10020
U.S.A.
Tel: (001)347-983-5186