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 ( X0 , u 0, λ 0) satisfies the equations Xu = λ u, null. We investigate the conditions under which unique differentiable functions λ( X) and u (X ) exist in a neighborhood of X 0 satisfying λ( X0 ) = λ O, u (X 0) = u 0, X = λ u, and null. 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.
Volume (Year): 1 (1985)
Issue (Month): 02 (August)
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