Geodesic
On melancholic magic squares
Discussiones Mathematicae. Probability and Statistics, Tome 33 (2013) no. 1-2, pp. 111-119

Voir la notice de l'article provenant de la source Library of Science

Starting with Dürer's magic square which appears in the well-known copper plate engraving Melencolia we consider the class of melancholic magic squares. Each member of this class exhibits the same 86 patterns of Dürer's magic square and is magic again. Special attention is paid to the eigenstructure of melancholic magic squares, their group inverse and their Moore-Penrose inverse. It is seen how the patterns of the original Dürer square to a large extent are passed down also to the inverses of the melancholic magic squares.
Keywords: magic squares, patterns, group inverse, Moore-Penrose inverse, eigenvalues and eigenvectors 
Trenkler, Götz; Trenkler, Dietrich. On melancholic magic squares. Discussiones Mathematicae. Probability and Statistics, Tome 33 (2013) no. 1-2, pp. 111-119. http://geodesic.mathdoc.fr/item/DMPS_2013_33_1-2_a7/
@article{DMPS_2013_33_1-2_a7,
     author = {Trenkler, G\"otz and Trenkler, Dietrich},
     title = {On melancholic magic squares},
     journal = {Discussiones Mathematicae. Probability and Statistics},
     pages = {111--119},
     year = {2013},
     volume = {33},
     number = {1-2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DMPS_2013_33_1-2_a7/}
}
TY  - JOUR
AU  - Trenkler, Götz
AU  - Trenkler, Dietrich
TI  - On melancholic magic squares
JO  - Discussiones Mathematicae. Probability and Statistics
PY  - 2013
SP  - 111
EP  - 119
VL  - 33
IS  - 1-2
UR  - http://geodesic.mathdoc.fr/item/DMPS_2013_33_1-2_a7/
LA  - en
ID  - DMPS_2013_33_1-2_a7
ER  - 
%0 Journal Article
%A Trenkler, Götz
%A Trenkler, Dietrich
%T On melancholic magic squares
%J Discussiones Mathematicae. Probability and Statistics
%D 2013
%P 111-119
%V 33
%N 1-2
%U http://geodesic.mathdoc.fr/item/DMPS_2013_33_1-2_a7/
%G en
%F DMPS_2013_33_1-2_a7

[1] W.S. Andrews, Magic Squares and Cubes (Dover, New York, 1960).

[2] A. Ben-Israel and T.N.E. Greville, Generalized Inverses: Theory and Applications (Springer, 2nd edition, 2003).

[3] H.D. Heinz and J.R. Hendricks, Magic Square Lexicon: Illustrated, HDH, Surrey, BC, 2000.

[4] G.P.H. Styan, G. Trenkler and K.L. Chu, An Illustrated Introduction to Yantra Magic Squares and Agrippa-type Magic Squares, Lectures on Matrix and Graph Methods, Ed. by R.B. Bapat, S. Kirkland, U.M. Prasad and S. Puntanen (Manipal University Press, 2012), 159-220.

[5] D. Trenkler and G. Trenkler, Magic squares, melancholy and the Moore-Penrose inverse, Image 27 (2001), 3-9.