Dot Products and Matrix Properties of 4×4 Strongly Magic Squares

Neeradha. C. K.; V. Madhukar Mallayya

doi:doi:10.11648/j.ijtam.20170302.13

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Dot Products and Matrix Properties of 4×4 Strongly Magic Squares

Neeradha. C. K., V. Madhukar Mallayya

Received: 4 November 2016 Accepted: 27 December 2016 Published: 13 February 2017

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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.

DOI	10.11648/j.ijtam.20170302.13
Published in	International Journal of Theoretical and Applied Mathematics (Volume 3, Issue 2, April 2017)
Page(s)	64-69
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Strongly Magic Square (SMS), Dot Products of SMS, Eigen Values of SMS, Rank and Determinant of SMS

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.

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APA Style

Neeradha. C. K., V. Madhukar Mallayya. (2017). Dot Products and Matrix Properties of 4×4 Strongly Magic Squares. International Journal of Theoretical and Applied Mathematics, 3(2), 64-69. https://doi.org/10.11648/j.ijtam.20170302.13

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ACS Style

Neeradha. C. K.; V. Madhukar Mallayya. Dot Products and Matrix Properties of 4×4 Strongly Magic Squares. Int. J. Theor. Appl. Math. 2017, 3(2), 64-69. doi: 10.11648/j.ijtam.20170302.13

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AMA Style

Neeradha. C. K., V. Madhukar Mallayya. Dot Products and Matrix Properties of 4×4 Strongly Magic Squares. Int J Theor Appl Math. 2017;3(2):64-69. doi: 10.11648/j.ijtam.20170302.13

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@article{10.11648/j.ijtam.20170302.13,
  author = {Neeradha. C. K. and V. Madhukar Mallayya},
  title = {Dot Products and Matrix Properties of 4×4 Strongly Magic Squares},
  journal = {International Journal of Theoretical and Applied Mathematics},
  volume = {3},
  number = {2},
  pages = {64-69},
  doi = {10.11648/j.ijtam.20170302.13},
  url = {https://doi.org/10.11648/j.ijtam.20170302.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20170302.13},
  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.},
 year = {2017}
}

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TY  - JOUR
T1  - Dot Products and Matrix Properties of 4×4 Strongly Magic Squares
AU  - Neeradha. C. K.
AU  - V. Madhukar Mallayya
Y1  - 2017/02/13
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DO  - 10.11648/j.ijtam.20170302.13
T2  - International Journal of Theoretical and Applied Mathematics
JF  - International Journal of Theoretical and Applied Mathematics
JO  - International Journal of Theoretical and Applied Mathematics
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AB  - 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.
VL  - 3
IS  - 2
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Author Information

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

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Plain Text BibTeX RIS

APA Style

Neeradha. C. K., V. Madhukar Mallayya. (2017). Dot Products and Matrix Properties of 4×4 Strongly Magic Squares. International Journal of Theoretical and Applied Mathematics, 3(2), 64-69. https://doi.org/10.11648/j.ijtam.20170302.13

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ACS Style

Neeradha. C. K.; V. Madhukar Mallayya. Dot Products and Matrix Properties of 4×4 Strongly Magic Squares. Int. J. Theor. Appl. Math. 2017, 3(2), 64-69. doi: 10.11648/j.ijtam.20170302.13

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AMA Style

Neeradha. C. K., V. Madhukar Mallayya. Dot Products and Matrix Properties of 4×4 Strongly Magic Squares. Int J Theor Appl Math. 2017;3(2):64-69. doi: 10.11648/j.ijtam.20170302.13

Copy | Download

@article{10.11648/j.ijtam.20170302.13,
  author = {Neeradha. C. K. and V. Madhukar Mallayya},
  title = {Dot Products and Matrix Properties of 4×4 Strongly Magic Squares},
  journal = {International Journal of Theoretical and Applied Mathematics},
  volume = {3},
  number = {2},
  pages = {64-69},
  doi = {10.11648/j.ijtam.20170302.13},
  url = {https://doi.org/10.11648/j.ijtam.20170302.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20170302.13},
  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.},
 year = {2017}
}

Copy | Download

TY  - JOUR
T1  - Dot Products and Matrix Properties of 4×4 Strongly Magic Squares
AU  - Neeradha. C. K.
AU  - V. Madhukar Mallayya
Y1  - 2017/02/13
PY  - 2017
N1  - https://doi.org/10.11648/j.ijtam.20170302.13
DO  - 10.11648/j.ijtam.20170302.13
T2  - International Journal of Theoretical and Applied Mathematics
JF  - International Journal of Theoretical and Applied Mathematics
JO  - International Journal of Theoretical and Applied Mathematics
SP  - 64
EP  - 69
PB  - Science Publishing Group
SN  - 2575-5080
UR  - https://doi.org/10.11648/j.ijtam.20170302.13
AB  - 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.
VL  - 3
IS  - 2
ER  -

Copy | Download