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Permutation matrices play an important role in understand the structure ofmagic squares. In this work, we use a class of symmetric permutation matricesthan can be used to categorize magic squares. Many magic squares with a highdegree of symmetry are studied, including classes that are generalizations ofthose categorized by Dudeney in 1917. We show that two classes of such magicsquares are singular and the eigenspectra of such magic squares are highlystructured. Lastly, we prove that natural magic squares of singly-even order ofthese classes do note exist.

The Magic of Permutation Matrices: Categorizing, Counting and Eigenspectra of Magic Squares