On commutativity of Discrete Fourier Transform

• In this paper we have studied the commutative properties of general DFT matrices UnUn. We find complete solutions for n=2,3,4,5n=2,3,4,5 theoretically.
• We reduced problem of solving n2n2 equations into solving two different systems of equations of n24 equations and same number of variables.
• We use the idea that set of symmetric and skew matrices form vector subspaces of space of matrices and space of matrices is the direct sum of the above two subspaces.

In this paper we have studied the commutative properties of general Discrete Fourier Transform (DFT) matrices UnUn. The problem is to characterize matrices AnAn that commute with UnUn. We find complete solutions for AnAn up to n=5n=5 theoretically. We also provide a major result towards the complete solutions for general n  . To find AnAn which commutes with UnUn one needs to solve a system of n2n2 linear equations of n2n2 variables. We reduced this problem into solving two different systems of linear equations of more or less n2/4n2/4 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.