Defect and equivalence of unitary matrices. The Fourier case

TL;DR

This paper studies the defect of unitary matrices, especially Fourier matrices, using a spectral approach, and explores how this defect behaves under Kronecker products and symmetries related to finite abelian groups.

Contribution

It provides a spectral characterization of the defect, formulas for Fourier matrices, and insights into symmetry and equivalence of Fourier matrices from finite abelian groups.

Findings

D(U) is the multiplicity of 1 in the spectrum of I_U

D(F) formulas for Fourier matrices of prime power size

D(U_1 x ... x U_r) multiplicativity under coprime sizes

Abstract

Consider the real space D_U of directions moving into which from a unitary N x N matrix U we do not disturb its unitarity and the moduli of its entries in the first order. dim( D_U ) is called the defect of U and denoted D(U). We give an account of Alexander Karabegov's theory where D_U is parametrized by the imaginary subspace of the eigenspace, associated with lambda = 1, of a certain unitary operator I_U on the N x N complex matrices, and where D(U) is the multiplicity of 1 in the spectrum of I_U. This characterization allows us to establish dependence of D(U_1 x ... x U_r) - where x stands for the Kronecker product - on D(U_k)'s, to derive formulas expressing D(F) for a Fourier matrix F of the size being a power of a prime number, as well as to show the multiplicativity of D(F) with respect to Kronecker factors of F if their sizes are pairwise relatively prime. Also partly due to the role of symmetries of U in the determination of the eigenspaces of I_U we study the 'permute and enphase' symmetries and equivalence of Fourier matrices, associated with arbitrary finite abelian groups.