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Upper Bounds on Eigenvalue Variation

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4 Author(s)
Guoxing Wu ; Dept. of Math., Northeast Forestry Univ., Harbin, China ; Yinyin Huang ; Duanmei Zhou ; Yanjun Yan

Let A and à = D1*AD2 be two n × n diagonalizable matrices with eigendecomposition A = XΛX-1 and A = X̃Λ̃X̃-1, where D1, D2, X and X̃ are nonsingular, and Λ = diag(λ1,⋯, λn) and Λ̃ = diag(λ̃1L ⋯, λ̃n). Li [1] proved that if λ1 ≥ λ2 ≥ ⋯ ≥ λn ≥ 0 and λ1 ≥ λ2 ≥ ⋯ ≥ λn ≥ 0, then max1 ≤ j ≤ nj-λ̃jj| ≤ ∥X-12∥X̃∥2∥D22 × ∥X̃-1 (D1*-D2-1)X∥2, max1≤ j ≤ nj-λ̃j/λ̃j| ≤ ∥X-12∥X̃∥2∥D1-*2 × ∥X̃-1(D1*-D2-1)X∥2. In this note, we show that the bounds are valid under slightly more general conditions.

Published in:

Computational Sciences and Optimization (CSO), 2011 Fourth International Joint Conference on

Date of Conference:

15-19 April 2011