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

Published in International Journal of Theoretical and Applied Mathematics (Volume 3, Issue 2)

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.

Published in	International Journal of Theoretical and Applied Mathematics (Volume 3, Issue 2)
DOI	10.11648/j.ijtam.20170302.13
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), 2017. 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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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
ER  -

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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