Magic square spectra

Abstract To study the eigenvalues of low order singular and non-singular magic squares we begin with some aspects of general square matrices. Additional properties follow for general semimagic squares (same row and column sums), with further properties for general magic squares (semimagic with same diagonal sums). Parameterizations of general magic squares for low orders are examined, including factorization of the linesum eigenvalue from the characteristic polynomial. For nth order natural magic squares with matrix elements 1 , … , n 2 we find examples of some remarkably singular cases. All cases of the regular (or associative, or symmetric) type (antipodal pair sum of 1 + n 2 ) with n - 1 zero eigenvalues have been found in the only complete sets of these squares (in fourth and fifth order). Both the Jordan form and singular value decomposition (SVD) have been useful in this study which examines examples up to 8th order. In fourth order these give examples illustrating a theorem by Mattingly that even order regular magic squares have a zero eigenvalue with odd algebraic multiplicity, m. We find 8 cases with m = 3 which have a non-diagonal Jordan form. The regular group of 48 squares is completed by 40 squares with m = 1 , which are diagonable. A surprise finding is that the eigenvalues of 16 fourth order pandiagonal magic squares alternate between m = 1 , diagonable, and m = 3 , non-diagonable, on rotation by π / 2 . Two 8th order natural magic squares, one regular and the other pandiagonal, are also examined, found to have m = 5 , and to be diagonable. Mattingly also proved that odd order regular magic squares have a zero eigenvalue with even multiplicity, m = 0 , 2 , 4 , . . . Analyzing results for natural fifth order magic squares from exact backtracking calculations we find 652 with m = 2 , and four with m = 4 . There are also 20, 604 singular seventh order natural ultramagic (simultaneously regular and pandiagonal) squares with m = 2 , demonstrating that the co-existence of regularity and pandiagonality permits singularity. The singular odd order examples studied are all non-diagonable.