酉不变范数
Unitarily invariant norms
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中文总结 AI 辅助
本综述研究n×n矩阵代数上的酉不变范数，构建相关理论框架并完成其 von Neumann 刻画，还探讨了 majorization 理论及矩阵分析相关关键结果。
中文摘要 AI 辅助
本综述论文对n×n矩阵代数上的酉不变范数进行了全面研究。该研究自然引出对称 gauge 函数理论，这是一类定义在ℝⁿ上、具有不变性和单调性的范数。我们通过考察绝对范数与单调范数并建立二者的等价性，构建了必要框架，从而为经典的关于 majorization 的 Hardy-Littlewood-Pólya 定理提供了新的见解。我们进一步通过 majorization 与 doubly stochastic 矩阵、凸性的联系，以及 Birkhoff 定理、Radó 定理和 König 关于项秩与行秩的定理等基础结果，深入探讨 majorization 理论。我们还研究了弱 majorization 并推导了一个刻画，该刻画在证明对称 gauge 函数的单调性中起着关键作用。在谱分析方面，我们回顾了矩阵分析中的关键结果，包括 Courant-Fischer 极小极大定理、Cauchy 交错定理、Ky Fan majorization 定理，以及任意矩阵奇异值的弱次可加性结果。这些内容最终形成了对酉不变范数的 von Neumann 刻画的详细证明，该证明为这类范数提供了完整且优雅的描述。文中还给出了一些说明性例子，以及作为应用的 Ky Fan 支配原理。
英文摘要
This survey paper provides a comprehensive study of unitarily invariant norms on the algebra of $n \times n$ matrices. This investigation leads naturally to the theory of symmetric gauge functions, a class of norms on $\mathbb{R}^n$ characterized by invariance and monotonicity properties. We develop the necessary framework by examining absolute and monotone norms and establishing their equivalence, thereby offering additional insight into the classical Hardy-Littlewood-Pólya theorem on majorization. The theory of majorization is further explored through its connections with doubly stochastic matrices, convexity, and fundamental results such as the Birkhoff and Radó theorems, as well as König's theorem on term rank and line rank. We also study weak majorization and derive a characterization that plays a crucial role in proving the monotonicity of symmetric gauge functions. On the spectral side, we review key results in matrix analysis, including the Courant-Fischer min-max theorem, the Cauchy interlacing theorem, and Ky Fan's majorization theorem, along with a weak subadditivity result for singular values of arbitrary matrices. These ingredients culminate in a detailed proof of von Neumann's characterization of unitarily invariant norms, which provides a complete and elegant description of this class of norms. Some illustrative examples, as well as the Ky Fan domination principle as an application, are also presented.
发表机构
- Pomona College（波莫纳学院）
- Université Laval（拉瓦尔大学）
- University of Agriculture（农业大学）
- University of Richmond（里士满大学）
机构由 AI 辅助整理，请以论文原文为准。
