A new derivation of a formula by Kato
| Autor: | Hristo S. Sendov, Brendan P. W. Ames |
|---|---|
| Rok vydání: | 2012 |
| Předmět: |
Power series
Pure mathematics Numerical Analysis Generalized inverse Algebra and Number Theory Directional derivative Symmetric matrix Eigenvalues Perturbation theory Analytic Kato Combinatorics Discrete Mathematics and Combinatorics Geometry and Topology Rellich Eigenvalues and eigenvectors Analytic function Resolvent Mathematics |
| Zdroj: | Linear Algebra and its Applications. 436(3):722-730 |
| ISSN: | 0024-3795 |
| DOI: | 10.1016/j.laa.2011.07.034 |
| Popis: | If the m th largest eigenvalue λ m ( A ) of a real symmetric matrix A is simple, then λ m ( · ) is an analytic function in a neighbourhood of A . In this note, we provide a new derivation of the classical formulae for the coefficients in the power series expansion of t ↦ λ m ( A + tE ) for any real symmetric matrix E and t close to 0. Kato’s classical derivation of that formula uses a complex-analytic approach involving properties of the resolvent of A + tE . Our derivation uses simple real-analytic and combinatorial arguments. In particular, we derive and utilize a formula for the derivative of the Moore–Penrose generalized inverse of the map X ↦ λ m ( X ) I - X in direction E at real symmetric matrix A for any real symmetric matrix E . |
| Databáze: | OpenAIRE |
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