On commutativity of Discrete Fourier Transform

In this paper we have studied the commutative properties of general Discrete Fourier Transform (DFT) matrices $U_n$ . The problem is to characterize matrices $A_n$ that commute with $U_n$ . We find complete solutions for $A_n$ up to $n=5$ theoretically. We also provide a major result towards the complete solutions for general n. To find $A_n$ which commutes with $U_n$ one needs to solve a system of $n^2$ linear equations of $n^2$ variables. We reduced this problem into solving two different systems of linear equations of more or less $n^2/4$ many variables and same number of equations. To do this reduction we use the idea of symmetric, skew symmetric matrices as well as we consider the set of matrices as a vector space and use direct sum of subspaces.