关于矩阵张量积数值半径的两个问题

Two problem of the numerical radius on tensor power of matrix

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).

This paper proves that for any n×n matrices A1,...,Ak∈L(U)we have r(A1...Ak)≥ki=1r(Ai); and for any two n×n matrices A,B∈L(U) we have r(AB)=r(BA); where r(A) denotes the numerical radius of A.And it shows that the inequalities r(AB)≤r(A)r(B) and r(AA)≤r2(A) do not hold in general, particularly, r(A1...Ak)=ks=1r(AS) for normal matrices A1,A2,...,Ak.