摘要
用H表示实数域上的四元数体.设a=a+bi+cj+dk∈H(a,b,c,d是实数),定义a的模为│a│=a^2+b^2+c^2+d^2(1/2).易知│αβ│=│α││β│,│α+β│≤│α│+│β│(α,β∈H).设a的共轭四元数为a=a—bi—cj—dk,显然│α~2│=aa=aa.用H^(m×n)表示H上的m×n矩阵的集合;用H_R^(n×n)表示可中心化的n阶四元数矩阵的集合;用SH^(n×n)表示n阶自共轭四元数矩阵的集合;用SH_≥^(n×n)表示n阶半正定自共轭四元数矩阵的集合;由[2]知SH^n×n_>(?)SH^n×n_≥(?)SH^n×n(?)H^n×n_R(?)H^n×n,
In this paper, we give accurate estimation of eigenvalues and singular values of A + B,C*AC and AB, where A, B and C are quaternions matrices. These results improve and generalze the results in [4] and [5]. We also obtainsum from i=1 to k(1/k)()λ_2+μ_n-k+i≤sum from i=1 to k,(1/k),λ_i(A+B)≤sum form i=1 to k.(1/k)(λ_i+μ_i),for k=1,…,n. Where A and B are self-conjugate quaternions matrices of order n, and λ_1≥…≥λ_n,μ_1≥μ_n,λ_1,(A + B)≥…≥λ_n(A+B) be the eigenvalues of A,B and A + B, respectively.
