The Magic of Permutation Matrices: Categorizing, Counting and Eigenspectra of Magic Squares
Staab, Peter, Charles Fisher, Maggio, Mark, Michael Andrade, Erin Farrell, Haley Schilling · arXiv (Cornell University) · 2010
Permutation matrices play an important role in understand the structure of magic squares. In this work, we use a class of symmetric permutation matrices than can be used to categorize magic squares. Many magic squares with a high degree of symmetry are studied, including classes that are generalizations of those categorized by Dudeney in 1917. We show that two classes of such magic squares are singular and the eigenspectra of such magic squares are highly structured. Lastly, we prove that natural magic squares of singly-even order of these classes do note exist.