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On Differentiating Eigenvalues and Eigenvectors

Published online by Cambridge University Press:  18 October 2010

Jan R. Magnus
Jan R. Magnus
Affiliation:
London School of Economics
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Abstract

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.

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Type
Articles
Information
Econometric Theory , Volume 1 , Issue 2 , August 1985 , pp. 179 - 191
Copyright
Copyright © Cambridge University Press 1985

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References

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