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On Differentiating Eigenvalues and Eigenvectors

Jan R. Magnus
- 01 Aug 1985 - 
- Vol. 1, Iss: 02, pp 179-191
TLDR
In this paper, the conditions under which unique differentiable functions λ(X) and u(X), respectively, exist in a neighborhood of a square matrix (complex or otherwise) satisfying the equations Xu = λu, λO, and Xu = ǫ, were investigated.
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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Tilburg University
On differentiating eigenvalues and eigenvectors
Magnus, J.R.
Published in:
Econometric Theory
Publication date:
1985
Link to publication in Tilburg University Research Portal
Citation for published version (APA):
Magnus, J. R. (1985). On differentiating eigenvalues and eigenvectors.
Econometric Theory
,
1
, 179-191.
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References
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Journal ArticleDOI

The Commutation Matrix: Some Properties and Applications

J.R. Magnus, +1 more
- 01 Mar 1979 - 
TL;DR: In this paper, the expectation and covariance matrix of the Wishart distribution are derived, where the expectation is derived from the expectation matrix of a square matrix containing only zeros and ones.
Posted Content

The commutation matrix: Some properties and applications

J.R. Magnus, +1 more
- 01 Jan 1979 - 
TL;DR: In this article, the expectation and covariance matrix of the Wishart distribution are derived, where the expectation is derived from the expectation matrix of a square matrix containing only zeros and ones.
Journal ArticleDOI

On eigenvalues of matrices dependent on a parameter

Peter Lancaster
- 01 Dec 1964 - 
Journal ArticleDOI

The Elimination Matrix: Some Lemmas and Applications

J.R. Magnus, +1 more
- 01 Dec 1980 - 
TL;DR: In this paper, two transformation matrices are introduced, L and D, which contain zero and unit elements only, and they are used for maximum likelihood estimation of the multivariate normal distribution, the evaluation of Jacobians of transformations with symmetric or lower triangular matrix arguments, and the solution of matrix equations.
Journal ArticleDOI

Derivatives of the characteristic root of a synmetric or a hermitian matrix with two applications in multivariate analysis

Nariaki Sugiura
- 01 Jan 1973 - 
TL;DR: In this article, the authors derived the derivative of the α th largest sharactsristic root of a symmetric matrix S = (srs) with respect to srs (r ≦ s) at S = A = diag(λ 1,…, λp) wher λ 1 ≧… ≧ λ p and λα is assumed to be simple.
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