Reference for Taylor expansion of eigenvalues via group inverse

Identify a published reference that presents the Taylor expansion of a simple eigenvalue of a differentiable matrix family using the group inverse and normalized left/right eigenvectors, including explicit formulas for the first three derivatives as in Proposition A.1.

Background

The paper provides explicit derivative formulas for a simple eigenvalue in terms of the group inverse, left and right eigenvectors, and matrix derivatives. Although the approach is classical, the authors state they do not know an explicit reference and therefore include complete details.

A documented source would help anchor these formulas in the matrix analysis literature.