Extremality of numerical radii of tensor products of matrices
| Autor: | Yueh Hua Lu, Hwa Long Gau |
|---|---|
| Rok vydání: | 2019 |
| Předmět: |
Numerical Analysis
Algebra and Number Theory Complex matrix 010102 general mathematics 010103 numerical & computational mathematics Radius Rank (differential topology) 01 natural sciences Combinatorics Matrix (mathematics) Tensor product Discrete Mathematics and Combinatorics Geometry and Topology 0101 mathematics Numerical range Operator norm Mathematics |
| Zdroj: | Linear Algebra and its Applications. 565:82-98 |
| ISSN: | 0024-3795 |
| DOI: | 10.1016/j.laa.2018.12.008 |
| Popis: | For n-by-n and m-by-m complex matrices A and B, respectively, it is known that the inequality w ( A ⊗ B ) ≤ ‖ A ‖ w ( B ) holds, where w ( ⋅ ) and ‖ ⋅ ‖ denote, respectively, the numerical radius and the operator norm of a matrix. In this paper, we consider when this becomes an equality. We show that the equality w ( A ⊗ B ) = ‖ A ‖ w ( B ) holds if and only if A and B have k-by-k compressions A 1 and B 1 , respectively, such that rank ( ‖ A ‖ 2 I k − A 1 ⁎ A 1 ) ≤ min θ ∈ R dim ker ( w ( B ) I k − Re ( e i θ B 1 ) ) . We also give some consequences of this result. In particular, we show that if rank B ≤ sup { k ∈ N : ‖ A k ‖ = ‖ A ‖ k } , then w ( A ⊗ B ) = ‖ A ‖ w ( B ) . |
| Databáze: | OpenAIRE |
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