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关于正规矩阵特征值的扰动

孙继广   

  1. 中国科学院计算中心
  • 出版日期:1984-03-14 发布日期:1984-03-14

孙继广. 关于正规矩阵特征值的扰动[J]. 计算数学, 1984, 6(3): 334-336.

ON THE PERTURBATION OF THE EIGENVALUES OF A NORMAL MATRIX

  1. Sun Ji-guang Computing Center, Academia Sinica
  • Online:1984-03-14 Published:1984-03-14
设N与A均为n×n正规矩阵,其特征值分别为{v_i}_(i=1)~n与{α_i}_(i=1)~n。Hoffman和Wielandt证明了:存在1,2,…,n的一个排列π(1),π(2),…,π(n),使得|| ||_F表示Frobenius范数。 当N为n×n Hermite矩阵,A为n×n可对称化矩阵,即存在非奇异矩阵Q=I+X,使得Q~(-1)AQ为Hermite矩阵时,Stewart证明了:如果N与A的特征值分别
Let N be a n×n normal matrix with eigenvalues v_1, v_2, …, v_n, and let A be an×n diagonalizable matrix, i. e., there is a nonsingular matrix Q such that Q~(-1)AQ=diag (α_1, α_2, …, α_n). It is proved that there exists a suitable permutation π of the set{1,2, …, n} such that(sum from i=1 to n |v_i-α_(π(i))|~2)~(1/2)≤||Q||_2||Q~(-1)||_2||A-N||_F,where || ||_2 denotes the spectral norm, and || ||_F the Frobenius norm.
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[1] A. J. Hoffman, H. W. Wielandt, The variation of the spectrum of a normal matrix, Duke Math. J., 20 (1953) , 37-39.
[2] W. Kahan, Spectra of nearly hermitian matrices, Proc. Amer. Math. Soc., 48 (1975) , 11-17.
[3] G. W. Stewart, A note on non-Hermitian perturlations of Hermitian matrices, CNA-41, AD-745006, 1972.
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