Defect and Equivalence of Unitary Matrices. The Fourier Case. Part I

Abstract

Consider the real space 𝔻U of directions one can move in from a unitary N × N matrix U without disturbing its unitarity and the moduli of its entries in the first order. dimℝ (𝔻U) is called the defect of U and denoted D(U). We give an account of Alexander Karabegov’s theory where 𝔻U is parametrized by the imaginary subspace of the eigenspace, associated with λ = 1, of a certain unitary operator IU on 𝕄N, and where D(U) is the multiplicity of 1 in the spectrum of IU. This characterization allows us to establish the dependence of D(U(1) ⊗ … ⊗U(r)) 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, 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 IU we study the ‘permute and enphase’ symmetries and the equivalence of Fourier matrices, associated with arbitrary finite abelian groups. This work is divided in two parts — the present one and the second appearing in the next issue of OSID [1].