Unitarily invariant norms
Abstract
This survey paper provides a comprehensive study of unitarily invariant norms on the algebra of n × n matrices. This investigation leads naturally to the theory of symmetric gauge functions, a class of norms on ℝn 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.
Related
- Clarkson--McCarthy type inequalities, part II: refinements, unitary orbits, unitarily invariant norms and ℓs--ℓs estimates
- Measure Convergence under Weak Domination for Unitarily Invariant Norms on Finite von Neumann Algebras
- Universal optimality of the double-centred matrix under unitarily invariant norms for dissimilarity data
- Short curves in homogeneous spaces of the unitary group for unitarily invariant norms
- A Sharp Unitarily Invariant Norm Bound for the Off-Diagonal Block Perturbation of a Hermitian Matrix
BibTeX
@misc{garcia2026unitarily,
title = {{Unitarily invariant norms}},
author = {Stephan Ramon Garcia and Javad Mashreghi and Marek Ptak and William T. Ross},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02837},
eprint = {2609.02837},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02837},
}
RIS
TY - UNPB
TI - Unitarily invariant norms
AU - Stephan Ramon Garcia
AU - Javad Mashreghi
AU - Marek Ptak
AU - William T. Ross
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.02837
UR - https://arxiv.org/abs/2609.02837
AB - This survey paper provides a comprehensive study of unitarily invariant norms on the algebra of n × n matrices. This investigation leads naturally to the theory of symmetric gauge functions, a class of norms on 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.
ER -