标题： 复圆流形（CCM）黎曼流形优化方程的完整推导 显示英文标题

标题： A Complete Derivation of Complex Circle Manifold (CCM) Riemannian manifold Optimization Equations

摘要： 在回顾了MIMO通信系统、相控阵波束成形、雷达和控制理论等应用中的流形优化技术后，我们观察到复数圆流形（CCM）被广泛使用，但其基础关系和方程在文献中缺乏严格且自包含的推导。 本文提供了CCM关键属性的系统且严格的证明，包括其切空间和黎曼梯度运算，并明确将其与现实世界的优化问题联系起来。 我们的工作旨在为应用CCM流形优化的研究人员和实践者提供一个统一的参考。 显示英文摘要

摘要： After reviewing manifold optimization techniques in applications like MIMO communication systems, phased array beamforming, radar, and control theory, we observed that the Complex Circle Manifold (CCM) is widely employed, yet its foundational relations and equations lack a rigorous, self-contained derivation in the literature. This paper provides a systematic and rigorous proof of CCM's key properties, including its tangent space and Riemannian gradient operations, with explicit connections to real-world optimization problems. Our work aims to serve as a unified reference for researchers and practitioners applying CCM Manifold Optimization.