矩阵乘积的特征值与奇异值不等式

INEQUALITIES FOR THE EIGENVALUES AND SINGULAR VALUES OF THE PRODUCT OF MATRICES

获得矩阵乘积的特征值与奇异值的不等式。(1 )设 A、B为非负定 Hermite阵 ,1≤ i1 <… <ik≤ l≤ n,则∏kt=1λl- it+1 (AB)≥ ∏kt=1λl- t+1 (A)λn- it+1 (B) ;∏kt=1λn- l+it(AB)≤ ∏kt=1λn- l+t(A)λit(B)。 (2 )设 A、B∈ Cn× n,1≤ i1 <… <ik≤ l≤ n,则∏kt=1σl- it+1 (AB)≥ ∏kt=1σl- t+1 (A)σn- it+1 (B) ;∏kt=1σn- l+it(AB)≤ ∏kt=1σn- l+t(A)σit(B)。

In this paper,the following inequalities for the eigenvalues and singular values of the product of matrices are obtained. (1)Let A and B be positive semidefinite Hermite matrices and 1≤i 1<...<i k≤l≤n ,then∏kt=1λ l-i t+1 (AB)≥∏kt=1λ l-t+1 (A)λ n-i t+1 (B); ∏kt=1λ n-l+i t (AB)≤∏kt=1λ n-l+t (A)λ i t (B)? (2)Let A?B∈C n×n ,and 1≤i 1<...<i k≤l≤n ,then∏kt=1σ l-i t+1 (AB)≥∏kt=1σ l-t+1 (A)σ n-i t+1 (B); ∏kt=1σ n-l+i t (AB)≤∏kt=1σ n-l+t (A)σ i t (B)?