关于Rayleigh商矩阵  

ON THE RAYLEIGH QUOTIENT MATRIX

作　　者：刘新国[1] 许雅各 

机构地区：[1]山东海洋学院应用数学系

出　　处：《计算数学》1990年第2期208-213,共6页Mathematica Numerica Sinica

摘　　要：§1.预备知识 在不加注明的情况下,本文沿用[3]中的记号. 设A为n×n矩阵,Q及?为n×m矩阵,而且m<n,Q_*Q=I_m,?~*?=I_m,I_m为m阶单位阵.如何界定矩阵?_~*A?与Q~*AQ的特征值的扰动? 上述问题出现于计算矩阵特征值的幂法、QR方法和同时迭代法中.Suppose A is an n×n Hermitian Matrix, Q and ? are n×m matrices, m<n,Q~*Q=I_m, Q~*Q=I_m, I_m is the m×m identity matrix, AQ=QA, A=diag (λ_1,…, λ_m), and μ_1,…,μ_m are the eigenvalues of matrix ?~*A?.In this paper, the following theorems are proved: Theorem 1. (∑from=1 to m(|λ_j-μ_π(j)|)~2)^(1/2)≤4∥A∥_2d_F^2(Q, ?)where ∥ ∥_2 is the 2-norm of a matrix, π(1),…, π(m) are a permutation of {1,2,…,m}, and d_F(Q, ?)=[tr(I_m-Q~*QQ~*?)]^(1/2).Theorem 2. Without loss of generality, suppose λ_1≥λ_2≥…≥λ_m and μ_1≥μ_2≥…≥μ_m. Then |λ_j-μ_j|≤4∥A∥_2 d_2~2(Q,?), j=1, 2,…m,where d_2(Q, ?) = [∥I_m-?~*QQ~*?∥_2]^(1/2).

关 键 词：雷利商矩阵 特征值 埃尔米特矩阵 

分 类 号：O241.6[理学—计算数学]