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 An 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 An 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. (C) 2015 Elsevier B.V. All rights reserved.