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α: calibrated so average coauthorship-adjusted count equals average raw count
Let X0 be a square matrix (complex or otherwise) and u0 a (normalized) eigenvector associated with an eigenvalue λo of X0, so that the triple (X0, u0, λ0) satisfies the equations Xu = λu, . We investigate the conditions under which unique differentiable functions λ(X) and u(X) exist in a neighborhood of X0 satisfying λ(X0) = λO, u(X0) = u0, Xu = λu, and . We obtain the first and second derivatives of λ(X) and the first derivative of u(X). Two alternative expressions for the first derivative of λ(X) are also presented.