A Study on Numerical Radii of Tensor Products and Hadamard Products of Matrices

Project Details


Let A=[aij] be an n-by-n complex matrix A, its numerical radius is w(A)=max{|<Ax,x>|∈C∶x∈C^n, ||x||=1}. Let B=[bij] be an m-by-m complex matrix, the tensor product A ⊗B of A and B is the (mn)-by-(mn) matrix [aijB]. If m=n, then the Hadamard product A○B of A and B is the n-by-n matrix [aijbij]. The main concern of this project is the relations between the numerical radius of A⊗B (resp., A○B) and those of $A$ and $B$. For one direction, we have the following inequality.) and those of $A$ and $B$. For one direction, we have the following inequality:w(A○B)≤w(A⊗B)≤||A||w(B).In this project, we want to obtain necessary and sufficient conditions for the equality w(A⊗B)=||A||w(B) (resp., w(A○B)=||A||w(B)) to hold. For each inequality, we have given theconjecture for the equality to hold, we will prove these conjectures as the purpose of this project.
Effective start/end date1/08/2031/07/21

UN Sustainable Development Goals

In 2015, UN member states agreed to 17 global Sustainable Development Goals (SDGs) to end poverty, protect the planet and ensure prosperity for all. This project contributes towards the following SDG(s):

  • SDG 1 - No Poverty
  • SDG 5 - Gender Equality
  • SDG 17 - Partnerships for the Goals


  • numerical range
  • numerical radius
  • tensor product
  • Hadamard product


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