Let <italic>X</italic><sub>0</sub> be a square matrix (complex or otherwise) and <italic>u</italic><sub>0</sub> a (normalized) eigenvector associated with an eigenvalue λ<sup>o</sup> of <italic>X</italic><sub>0</sub>, so that the triple (<italic>X</italic><sub>0</sub>, <italic>u</italic><sub>0</sub>, λ<sub>0</sub>) satisfies the equations <italic>Xu</italic> = λ<italic>u</italic>, null. We investigate the conditions under which unique differentiable functions λ(<italic>X</italic>) and <italic>u</italic>(<italic>X</italic>) exist in a neighborhood of <italic>X</italic><sub>0</sub> satisfying λ(<italic>X</italic><sub>0</sub>) = λ<sub>O</sub>, <italic>u</italic>(<italic>X</italic><sub>0</sub>) = <italic>u</italic><sub>0</sub>, <italic>X</italic> = λ<italic>u</italic>, and null. We obtain the first and second derivatives of λ(<italic>X</italic>) and the first derivative of <italic>u</italic>(<italic>X</italic>). Two alternative expressions for the first derivative of λ(<italic>X</italic>) are also presented.
