令 A=[aij] 為n乘n 矩陣。而矩陣A的數值半徑其定義為w(A)=sup{|<Ax,x>|∈C∶x∈C^n, ||x||=1}. 令 B=[bij] 為m乘m 矩陣，則A和B的張量積為A ⊗B=[aijB]。若m=n，則A和B的哈達瑪積為A○B=[aijbij]。關於這兩個乘積的數值半徑，我們已知有以下的不等式：w(A○B)≤w(A⊗B)≤||A||w(B).本計畫的內容在於考慮上述不等式何時其等式會成立？相當於我們對下列不等式w(A⊗B)≤||A||w(B) 和 w(A○B)≤||A||w(B), 想找出當其等式成立時的充分必要條件為何。在計畫內容中，我們對於每一個等式成立的充分必要條件都給出了猜測，此計畫的目的在於證明這些猜測都是對的。 ;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.
