四元数矩阵的奇异值的几个不等式

SOME INEGUALITIES OF SINGULAR VALUES FOR QUATERNION MATRICES

本文证明了长方四元数矩阵奇异值的一些不等式:设H为四元数体,A∈H^(n×m),B∈H^(m×k),S=min{n,k},1≤l≤s,则 sum from i=1 to l σ_i(AB)≤sum from i=1 to l σ_i(A)σ_i(B) (ⅰ) sum from i=1 to l σ_s _(i+1)(AB)≥sum from i+j=m+s-l+1 σ_i(A)σ_j(B) (ⅱ) multiply from i=1 to l σ_i(A)σ_(m-i+1) (B)≤multiply from i=1 to l σ_i(AB)≤multiply from i=1 to l σ_i(A)σ_i(B) (ⅲ) 其中,σ_1(A)≥σ_2(A)≥…≥σ_m(A)≥0是A的从大到小的奇异值,当i>m时,σ_1(A)(?)0。不等式(ⅰ),(ⅱ),(ⅲ)包含或加强了文[3]、[4]、[5]的一些基本结果。

In this paper, We give some inegualities on the singular values of the product of several quaternion matrices. Let H be guaternion field, A∈H^(n×m), B∈H^(m×k), s(?)min {n, k},1≤l≤s, then sum from i=1 to l σ_i(AB)≤sum from i=1 to l σ_i(A)σ_i(B) sum from i=1 to l σ_(s-i+1)(AB)≥sum from i+j=m+s-l+1 σ_i(A)σ_j(B) multiply from i=1 to l σ_i(A)σ_(m-i+1) (B)≤multiply from i=1 to l σ_i(AB)≤multiply from i=1 to l σ_i(A)σ_i(B) where σ_1(A)≥σ_2(A)≥…≥σ_m(A)≥0 are singwlar values of A, and σ_1(A)(?)0 for i>m.