Asymptotic expansions of the ordered spectrum of symmetric matrices

Abstract

In this work, we build on ideas of Torki (2001 [6] ) and show that if a symmetric matrix-valued map t ↦ A ( t ) has a one-sided asymptotic expansion at t = 0 + of order K then so does t ↦ λ m ( A ( t ) ) , where λ m is the m th 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 λ m ( A + t E ) for any fixed symmetric matrices A and E .