Convexity and Star-shapedness of Real Linear Images of Special Orthogonal Orbits
| Autor: | Pan-Shun Lau, Nam-Kiu Tsing, Tuen-Wai Ng |
|---|---|
| Rok vydání: | 2016 |
| Předmět: |
Numerical Analysis
Algebra and Number Theory 15A04 15A18 Image (category theory) 010102 general mathematics 010103 numerical & computational mathematics Star (graph theory) 01 natural sciences Convexity Functional Analysis (math.FA) Mathematics - Functional Analysis Combinatorics Linear map Singular value FOS: Mathematics Discrete Mathematics and Combinatorics Geometry and Topology Orthogonal matrix 0101 mathematics Mathematics |
| DOI: | 10.48550/arxiv.1608.06101 |
| Popis: | Let A ∈ R n × n and SO n : = { U ∈ R n × n : U U t = I n , det U > 0 } be the set of n × n special orthogonal matrices. Define the (real) special orthogonal orbit of A by O ( A ) : = { U A V : U , V ∈ SO n } . In this paper, we show that the linear image of O ( A ) is star-shaped with respect to the origin for arbitrary linear maps L : R n × n → R l if n ≥ 2 l − 1 . In particular, for linear maps L : R n × n → R 2 and when A has distinct singular values, we study B ∈ O ( A ) such that L ( B ) is a boundary point of L ( O ( A ) ) . This gives an alternative proof of a result by Li and Tam on the convexity of L ( O ( A ) ) for linear maps L : R n × n → R 2 . |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|