Asymptotic expansions of the ordered spectrum of symmetric matrices

In this work, we build on ideas of Torki (2001 [6]) and show that if a symmetric matrix-valued map t↦A(t)t↦A(t) has a one-sided asymptotic expansion at t=0+t=0+ of order KK then so does t↦λm(A(t))t↦λm(A(t)), where λmλm is the mmth largest eigenvalue. We derive formulas for computing the coefficients A0,A1,…,AKA0,A1,…,AK 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 λm(A+tE)λm(A+tE) for any fixed symmetric matrices AA and EE.