A reduction algorithm for matrices depending on a parameter
ISSAC '99 Proceedings of the 1999 international symposium on Symbolic and algebraic computation
An algorithm for the eigenvalue perturbation problem: reduction of a κ-matrix to a Lidskii matrix
ISSAC '00 Proceedings of the 2000 international symposium on Symbolic and algebraic computation
A reduced form for perturbed matrix polynomials
Proceedings of the 2002 international symposium on Symbolic and algebraic computation
About Hölder condition numbers and the stratification diagram for defective eigenvalues
Computational science, mathematics and software
On matrix perturbations with minimal leading Jordan structure
Journal of Computational and Applied Mathematics - Special issue: Proceedings of the international conference on linear algebra and arithmetic, Rabat, Morocco, 28-31 May 2001
Perturbations of Jordan matrices
Journal of Approximation Theory
Eigenvalue perturbation bounds for Hermitian block tridiagonal matrices
Applied Numerical Mathematics
Structured Pseudospectra and the Condition of a Nonderogatory Eigenvalue
SIAM Journal on Matrix Analysis and Applications
On Explicit Recursive Formulas in the Spectral Perturbation Analysis of a Jordan Block
SIAM Journal on Matrix Analysis and Applications
The dynamics of matrix coupling with an application to krylov methods
NAA'04 Proceedings of the Third international conference on Numerical Analysis and its Applications
Structured Pseudospectra for Small Perturbations
SIAM Journal on Matrix Analysis and Applications
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Let A be a complex matrix with arbitrary Jordan structure and $\lambda$ an eigenvalue of A whose largest Jordan block has size n. We review previous results due to Lidskii [U.S.S. R. Comput. Math. and Math. Phys., 1 (1965), pp. 73--85], showing that the splitting of $\lambda$ under a small perturbation of A of order $\varepsilon$ is, generically, of order $\varepsilon^{1/n}$. Explicit formulas for the leading coefficients are obtained, involving the perturbation matrix and the eigenvectors of A. We also present an alternative proof of Lidskii's main theorem, based on the use of the Newton diagram. This approach clarifies certain difficulties which arise in the nongeneric case and leads, in some situations, to the extension of Lidskii's results. These results suggest a new notion of Hölder condition number for multiple eigenvalues, depending only on the associated left and right eigenvectors, appropriately normalized, not on the Jordan vectors.