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Power iteration with normalisation and Rayleigh-quotient eigenvalue estimates
Definition
Let and let be nonzero. The power iteration generated by is the sequence of normalised vectors
defined whenever .
When the iterates are used with a self-adjoint or Hermitian matrix, the associated Rayleigh-quotient eigenvalue estimates are
For , every defined iterate is normalised to unit length, and then . The same simplification holds at when is chosen to have unit length.
Used by
- Equal dominant moduli can make power iteration alternate or cycle instead of converging Counterexample
- Inverse iteration and shifted inverse iteration Definition
- For Hermitian matrices, the Rayleigh quotient and residual converge with the expected rates along power iteration Proposition
- If a diagonalisable matrix has a simple eigenvalue of strictly largest modulus and the start vector has a nonzero component in that eigendirection, power iteration converges projectively at the eigenvalue-ratio rate Theorem
Dependency tree · 0 levels
Nothing. This result depends on no other item in the library.
Sources
- Andrew Stuart and Jochen Voss, Matrix Analysis and Algorithms (standard reference, not scraped)