Unitarily invariant norms

Abstract

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.