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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.

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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