Spectra of order-4 special magic squares

A new parameterization for order-4 regular (or associative) magic square matrices leads to general formulas for their eigenvalues, eigenvectors, and singular value decomposition. Known transformations extend these results to order-4 pandiagonal and bent-diagonal magic squares. The effect of various transformations on the eigenvalues and singular values of these special magic squares is considered. Numerical examples are presented and numerical values are obtained from simple formulas for the eigenvalues and singular values of each of the 48 natural pandiagonal, regular, and bent-diagonal magic squares of order 4 and their reflections.