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) 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 mth largest eigenvalue. We derive formulas for computing the coefficients A₀,A₁,…,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+tE) for any fixed symmetric matrices A and E.