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Linear Normal Forms

  • James Murdock
Chapter
  • 536 Downloads
Part of the Springer Monographs in Mathematics book series (SMM)

Abstract

Consider a smooth (real or complex) matrix-valued function A(ε) of a (real) small parameter ε, having formal power series
$$ A\left( \varepsilon \right)\, \sim \,{A_{{0\,}}} + \varepsilon {A_1} + {\varepsilon^2}{A_2} + .... $$
(3.1.1)
How do the eigenvectors (or generalized eigenvectors) and eigenvalues of such a matrix vary with ε? This question arises, for instance, in studying the stability of the linear system of differential equations ẋ = A(ε)x. Or the nonlinear system ẋ = f(x, ε) may have a rest point x*(e) whose stability depends on the matrix A(ε) = fx(x*(ε), ε). The same question arises for diíferent reasons in quantum mechanics; in this case A(ε) is Hermitian, hence diagonalizable, and the interest focuses on the splitting (for ε ≠ 0) of eigenvalues that are equal when ε = 0. (These eigenvalues can be, for example, the spectral lines of an atom, which can split in the presence of an external field.)

Keywords

Normal Form Jordan Block Jordan Form Homological Equation Stripe Structure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag New York, Inc. 2003

Authors and Affiliations

  • James Murdock
    • 1
  1. 1.Mathematics DepartmentIowa State UniversityAmesUSA

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