On properties of special magic square matrices

By treating regular (or associative), pandiagonal, and most-perfect (MP) magic squares as matrices, we find a number of interesting properties and relationships. In addition, we introduce a new class of quasi-regular (QR) magic squares which includes regular and MP magic squares. These four classes of magic squares are called “special”.We prove that QR magic squares have signed pairs of eigenvalues just as do regular magic squares according to a well-known theorem of Mattingly. This leads to the fact that odd powers of QR magic squares are magic squares which also can be established directly from the QR condition. Since all pandiagonal magic squares of order 4 are MP, they are QR. Also, we show that all pandiagonal magic squares of order 5 are QR but higher-order ones may or may not be. In addition, we prove that odd powers of MP magic squares are MP. A simple proof is given of the known result that natural (or classic) pandiagonal and regular magic squares of singly-even order do not exist.We consider the reflection of a regular magic square about its horizontal or vertical centerline and prove that signed pairs of eigenvalues of the reflected square differ from those of the original square by the factor i. A similar result is found for MP magic squares and a subclass of QR magic squares.The paper begins with mathematical definitions of the special magic squares. Then, a number of useful matrix transformations between them are presented. Next, following a brief summary of the spectral analysis of matrices, the spectra of these special magic squares are considered and the results mentioned above are established. A few numerical examples are presented to illustrate our results.