A new derivation of a formula by Kato

If the mth largest eigenvalue λm(A) of a real symmetric matrix A is simple, then λm(·) is an analytic function in a neighbourhood of A. In this note, we provide a new derivation of the classical formulae for the coefficients in the power series expansion of t↦λm(A+tE) for any real symmetric matrix E and t close to 0. Kato’s classical derivation of that formula uses a complex-analytic approach involving properties of the resolvent of A+tE. Our derivation uses simple real-analytic and combinatorial arguments. In particular, we derive and utilize a formula for the derivative of the Moore–Penrose generalized inverse of the map X↦λm(X)I-X in direction E at real symmetric matrix A for any real symmetric matrix E.