Closest Normal Matrix Found Again Using Riemannian Optimization
Abstract
We propose an approach based on Riemannian optimization to compute a nearest normal matrix to a given one. The problem can be formulated as the minimization of a smooth function either on the manifold U(n) of unitary matrices of size n or on the flag manifold U (n)/U (1)n. The flag manifold is particularly suitable for theoretical analysis; we characterize the global maximum of the objective function and prove that, for generic inputs, its local minimizers are finitely many and isolated; in turn, this implies the original nearest normal matrix problem generically has finitely many local minimizers, all with distinct eigenvalues. We also develop a Riemannian trust-region method that improves substantially on classical algorithms and can handle considerably larger matrices, as well as a variant for computing the nearest real normal matrix. The paper is complemented by extensive numerical experiments.
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BibTeX
@misc{noferini2026closest,
title = {{Closest Normal Matrix Found Again Using Riemannian Optimization}},
author = {Vanni Noferini and Matvei Zhukov},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.28545},
eprint = {2608.28545},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.28545},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Closest Normal Matrix Found Again Using Riemannian Optimization
AU - Vanni Noferini
AU - Matvei Zhukov
PY - 2026
DA - 2026/08/28
PB - arXiv
DO - 10.48550/arxiv.2608.28545
UR - https://arxiv.org/abs/2608.28545
AB - We propose an approach based on Riemannian optimization to compute a nearest normal matrix to a given one. The problem can be formulated as the minimization of a smooth function either on the manifold U(n) of unitary matrices of size n or on the flag manifold U (n)/U (1)^n. The flag manifold is particularly suitable for theoretical analysis; we characterize the global maximum of the objective function and prove that, for generic inputs, its local minimizers are finitely many and isolated; in turn, this implies the original nearest normal matrix problem generically has finitely many local minimizers, all with distinct eigenvalues. We also develop a Riemannian trust-region method that improves substantially on classical algorithms and can handle considerably larger matrices, as well as a variant for computing the nearest real normal matrix. The paper is complemented by extensive numerical experiments.
ER -