[Paper Review] The Magic of Permutation Matrices: Categorizing, Counting and Eigenspectra of Magic Squares

Peter Staab, Charles Fisher+4 more|arXiv (Cornell University)|Jul 19, 2010

Graph Labeling and Dimension Problems7 references3 citations

TL;DR

This paper generalizes the role of permutation matrices in categorizing, counting, and analyzing the eigenspectra of magic squares, particularly extending Mattingly's result that even-sized regular magic squares are singular. It introduces new classes of magic squares (types A and B) and proves that all even-ordered type A and B magic squares are singular, with detailed analysis of their eigenvalues and symmetries under bisymmetric and 90°-symmetric permutation matrices.

ABSTRACT

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.

Motivation & Objective

• To extend Dudeney’s classification of 4×4 magic squares to higher-order magic squares using permutation matrices.
• To generalize Mattingly’s result that all even-ordered regular magic squares are singular to broader classes of magic squares.
• To analyze the eigenspectra of magic squares under transformations by bisymmetric and 90°-symmetric permutation matrices.
• To establish that type A and B magic squares of even order are singular, with specific eigenvalue properties.
• To provide a unified framework for understanding symmetries and spectral properties of magic squares using matrix algebra and permutation group theory.

Proposed method

• Uses matrix definitions with standard basis vectors e_i, all-ones vector e, and reverse matrix J to formalize magic squares and semi-magic squares.
• Defines magic squares via row, column, main diagonal, and anti-diagonal sum constraints using matrix equations (7)–(10).
• Introduces bisymmetric and 90°-symmetric permutation matrices as key tools to generate families of equivalent magic squares under transformation.
• Applies similarity transformations (PAP^T) and conjugation with J to preserve eigenspectra, proving invariance under these symmetries.
• Employs rank analysis and row reduction (e.g., subtracting rows) to determine nullity and confirm singularity of magic squares.
• Uses numerical computation to determine specific eigenspectra, such as for a 7×7 magic square with eigenvalues including 260 and complex conjugate pairs.

Experimental results

Research questions

• RQ1 How can permutation matrices be used to systematically categorize and generate families of magic squares beyond simple rotations and reflections?
• RQ2 Are there higher-order generalizations of Dudeney’s 4×4 magic square types that retain structural and spectral properties?
• RQ3 What is the determinant and eigenspectrum of even-ordered magic squares under generalized type A and B classifications?
• RQ4 Do bisymmetric and 90°-symmetric permutation matrices preserve the eigenspectrum of magic squares under transformation?
• RQ5 Can the singularity of even-ordered magic squares be proven for broader classes than just regular (type III) squares?

Key findings

• All even-ordered type A and B magic squares are singular, generalizing Mattingly’s result on regular magic squares.
• The eigenspectrum of a 7×7 magic square includes eigenvalues {260, 0, 0, 0, -53.8553, 49.6710, 2.0921±43.6941i}, with rank 5 and nullity 2.
• Bisymmetric permutation matrices preserve the eigenspectrum under conjugation, and transformations like PAP^T maintain spectral invariance.
• The reverse matrix J and 90°-symmetric permutations generate new magic squares with related eigenspectra, forming distinct spectral families.
• The transformation J P_i A P_i has the same eigenspectrum as P_i A P_i, and their transposes preserve spectral properties.
• There are no natural singly-even regular magic squares, and this extends to type A magic squares, confirming structural limitations in even-order cases.