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].

中文翻译：

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酉矩阵的缺陷和等价性。傅立叶案例。第一部分

考虑真实空间𝔻ü一个方向可以从一个单一的 N × N 矩阵 U 移动而不会干扰它的单一性和它的一阶项的模数。暗淡ℝ（𝔻ü) 称为 U 的缺陷，记为 D(U)。我们给出了 Alexander Karabegov 的理论，其中 𝔻ü由某个酉算子 I 的与 λ = 1 相关联的本征空间的虚子空间参数化ü在𝕄ñ, 其中 D(U) 是 I 的频谱中 1 的重数ü. 这种表征使我们能够建立 D(U(1)⊗ … ⊗U(r)) 在 D(U(k)) 的公式，以导出表示大小为素数幂的傅立叶矩阵 F 的 D(F) 的公式，以及显示 D(F) 相对于 F 的克罗内克因子的乘法性，如果它们的大小是成对相对素数。也部分是由于 U 的对称性在确定 I 的特征空间中的作用ü我们研究了与任意有限阿贝尔群相关的傅里​​叶矩阵的“置换和相位”对称性和等价性。这项工作分为两部分——目前的一个和第二个出现在下一期 OSID [1] 中。