Asymptotic expansions of the ordered spectrum of symmetric matrices

作者：Ames Brendan P W; Sendov Hristo S^*

来源：Nonlinear Analysis-Theory Methods & Applications, 2010, 72(11): 4288-4297.

DOI：10.1016/j.na.2010.02.005

• Eigenvalues
• Symmetric matrix
• Perturbation theory
• Asymptotic
• Analytic
• Rellich
• Kato

In this work, we build on ideas of Torki (2001 [6]) and show that if a symmetric matrix-valued map t bar right arrow A(t) has a one-sided asymptotic expansion at t = 0(+) of order K then so does t bar right arrow lambda(m)(A(t)), where lambda(m) is the mth largest eigenvalue. We derive formulas for computing the coefficients A(0), A(1),..., A(k) in the asymptotic expansion. As an application of the approach we give a new proof of a classical result due to Kato (1976 [3]) about the one-sided analyticity of the ordered spectrum under analytic perturbations. Finally, as a demonstration of the derived formulas, we compute the first three terms in the asymptotic expansion of lambda(m) (A + tE) for any fixed symmetric matrices A and E.

• 出版日期2010-6-1