Extremality of numerical radii of tensor products of matrices

Hwa Long Gau, Yueh Hua Lu

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


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 A1 and B1, respectively, such that rank(‖A‖2Ik−A1 A1)≤minθ∈R⁡dim⁡ker⁡(w(B)Ik−Re(eB1)). We also give some consequences of this result. In particular, we show that if rankB≤sup⁡{k∈N:‖Ak‖=‖A‖k}, then w(A⊗B)=‖A‖w(B).

Original languageEnglish
Pages (from-to)82-98
Number of pages17
JournalLinear Algebra and Its Applications
StatePublished - 15 Mar 2019


  • Numerical radius
  • Numerical range
  • Tensor product


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