关于四元数矩阵乘积迹的不等式

Inequalities on Trace of Product of Quaternions Matrices

设 H^(m×n)为 m×n 四元数矩阵的集合,σ_1(A)≥…≥σ_n(A)为 A∈H^(mxn)的奇异值。本文证明了:1)设 A∈H^(mxm),B∈H^(mxm),r=min(m,m),则|tr(4B)|≤c r σ_i(A)σ_i(B).2)设 A_i∈H^(mxm),i=1,2,…,n,(A_1A_2…A_n)k为 A_1A_2…A_n 的任一个 k 阶主子阵,则|tr(A_1.A_2…A_n)_k|≤sun form i=1 to k σ_i(A_1)…σ_i(A_n).我们还得到四元数矩阵迹的其它一些不等式。这些结果推广和改进了文[1],[2]中的结果,进一步解决了 Bellman 猜想。

Let Hm×ndenote the set of all m×n quaternions matrices,let δ1（A）≥… ≥δn,（A） be the singular values of A∈Hm×n.This paper proved:1)If A∈ Hm×n,B∈Hm×n,r=min（m,n）,then |tr（AB）|≤sum from i=1 to r δi（A）δi（B）.2)If Ai∈ Hm×n,i=1,2,…,n,let（A1A2…An）k denote the principal submatrix of order k of A1A2…An,then |tr（A1A2…An）k|≤sum from i=1 to k δi（AI）…δi（An）.We also also obtain other inepualities of trace for quaternions matrices.These results generalize and improve the results in [1],[2],and solve the Bellman’s guess.