Finding the Unique Permutation Matrix for Reverse Order Kronecker Product Intuitively

Abstract

With the adverse research in field of engineering and technology, the use of Kronecker product highlighted from many aspects by researchers in different fields of science, especially considering digital signal processing where Discrete Fourier Transform plays a supreme role. This transform can be performed using Kronecker product in an iterative way. The implementation of DFT using this product needs a reverse order multiplication that can easily be catered for by applying few techniques of linear algebra i.e. by using a unique permutation matrix, on the original Kronecker Product. The permutation matrix obtained from “I” interchanging of rows / columns from an identity matrix. A total of n! permutation matrices will be obtained using this methodology. This paper presents a technique for finding the unique permutation matrix out of n! matrices. This uniquely identified permutation matrix is used to attain the reverse order Kronecker product without using the same technique, as used for obtaining the original Kronecker product.

Keywords Discrete fourier transform (DFT), Digital signal processing (DSP), Intuitive method, Kronecker product (KP), Permutation matrix (PM)