On Differentiating Eigenvalues and Eigenvectors

Let X 0 be a square matrix (complex or otherwise) and u 0 a (normalized) eigenvector associated with an eigenvalue λ o of X 0 , so that the triple ( X 0 , u 0 , λ 0 ) satisfies the equations Xu = λ u , . We investigate the conditions under which unique differentiable functions λ( X ) and u ( X ) exist in a neighborhood of X 0 satisfying λ( X 0 ) = λ O , u ( X 0 ) = u 0 , X u = λ u , 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.