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 characterisation 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 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 ℐU we study the ‘permute and enphase’ symmetries and equivalence of Fourier matrices, associated with arbitrary finite abelian groups. This work is published as two papers — the first part [1] and the current second one.

中文翻译：

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酉矩阵的缺陷和等价性。傅立叶案例。第二部分

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构子空间参数化ü在𝕄ñ, 其中 D(U) 是 1 在 ℐ 的谱中的重数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。这项工作作为两篇论文发表——第一部分 [1] 和当前的第二部分。