J. Magnus. Econometric Theory, 1 (2):
pp. 179-191(1985)
Abstract
Let X0 be a square matrix (complex or otherwise) and u0 a (normalized) eigenvector associated with an eigenvalue λ 0 of X0, so that the triple (X0,u0,λ 0) satisfies the equations Xu = λu, u0 * u=1. We investigate the conditions under which unique differentiable functions λ(X) and u(X) exist in a neighborhood of X0 satisfying λ (X0)=λ 0,u(X0)=Xu=λ u, and u0 * u=1. 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.
%0 Journal Article
%1 magnus1985differentiating
%A Magnus, Jan R.
%D 1985
%I Cambridge University Press
%J Econometric Theory
%K eigenvalues eigenvectors linear_algebra matrix_differentiation sensitivity_analysis
%N 2
%P pp. 179-191
%T On Differentiating Eigenvalues and Eigenvectors
%U http://www.jstor.org/stable/3532409
%V 1
%X Let X0 be a square matrix (complex or otherwise) and u0 a (normalized) eigenvector associated with an eigenvalue λ 0 of X0, so that the triple (X0,u0,λ 0) satisfies the equations Xu = λu, u0 * u=1. We investigate the conditions under which unique differentiable functions λ(X) and u(X) exist in a neighborhood of X0 satisfying λ (X0)=λ 0,u(X0)=Xu=λ u, and u0 * u=1. 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.
@article{magnus1985differentiating,
abstract = {Let X0 be a square matrix (complex or otherwise) and u0 a (normalized) eigenvector associated with an eigenvalue λ 0 of X0, so that the triple (X0,u0,λ 0) satisfies the equations Xu = λu, u0 * u=1. We investigate the conditions under which unique differentiable functions λ(X) and u(X) exist in a neighborhood of X0 satisfying λ (X0)=λ 0,u(X0)=Xu=λ u, and u0 * u=1. 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.},
added-at = {2015-10-25T06:04:48.000+0100},
author = {Magnus, Jan R.},
biburl = {https://www.bibsonomy.org/bibtex/2aea2729487cb22971bf73677bfb2f2ca/peter.ralph},
interhash = {46920ec37bdf6762871d37ba29b3a7c4},
intrahash = {aea2729487cb22971bf73677bfb2f2ca},
journal = {Econometric Theory},
keywords = {eigenvalues eigenvectors linear_algebra matrix_differentiation sensitivity_analysis},
language = {English},
number = 2,
pages = {pp. 179-191},
publisher = {Cambridge University Press},
timestamp = {2015-10-25T06:04:48.000+0100},
title = {On Differentiating Eigenvalues and Eigenvectors},
url = {http://www.jstor.org/stable/3532409},
volume = 1,
year = 1985
}