Convex combinations of three real orthogonal matrices
Year: 2026
Type: preprint
Abstract: Pang proved that every real square matrix is a linear combination of three orthogonal matrices. Adapting his rank-one perturbation construction, we show that every real square contraction is a convex combination of three orthogonal matrices, with two equal coefficients. Three terms are necessary in every dimension at least two, so the real operator-norm unit ball has Carath\'eodory number three. Scaling gives a three-term representation whose absolute coefficients sum to the operator norm.For convex combinations of two orthogonal matrices, best approximation in any orthogonally invariant norm is attained at one of two explicit spectral candidates. This gives sharp approximation radii and lower bounds for the coefficients of three-term decompositions. In dimension three, we characterize all prescribed triples of convex weights, including factors of different determinants, degenerate spectra and zero weights. Finally, we record a consequence of the Fan--Pall compression theorem: every real contraction with more rows than columns is the arithmetic mean of two matrices with orthonormal columns.
Author: Nicolas, Jérôme
Language: English
Topic: Matrix Theory and Algorithms
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7220705457