矩阵张量积数值半径的一个不等式和一个等式

摘要

By using the properties of tensor product and numerical radius of matrix,for any n×n matrices A1,…,Ak∈L(U),the inequalityand for any two n×n matrices A,B∈L(U),the equality r(A(×)B)=r(B(×)A) is certainly hold are proved.Meanwhile the inequalities r(A(×)B)≤r(A)r(B) and r(A(×)A)≤r2(A) are all disappointingly incorrect through a simple illustration is demonstrated.%借助矩阵张量积和矩阵数值半径的性质,证明了不等式r(A1(×)…(×)Ak)≥∏ki=1r(Ai)和等式r(A(×)B)=r(B(×)A),其中A1,…,Ak,A,B∈L(U).同时,举例说明了不等式r(k(×)A)≤rk(A)不成立.而当A1,…,Ak为正规阵时,有r(A1(×)…(×)Ak)=∏ks=1r(As).