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## Eigenvalue Analysis

Summary:
• Motivation: PC are new variables, uncorrelated built from the old ones.
• Coefficients are obtained through eigenvalues of variance-covariance or correlation matrix.
• Eigenvalues represent variance explained.
• Also useful discriminant analysis, canonical variate analysis.
• Generalized eigenvalue problem .
• Statistics symmetrical eigenvalue problems.
• and same eigenvalues tranforms 's structure.
• similarity.
• Power method: simple non .
• .

• Advantageous for , easy to compute, only provides .
• orthogonal columns, re-orthogonalizing at each step.
• .
• Rate .
• -algorithm         where .
• Improvement 1: start with tridiagonal (Householder).
• Tridiagonal form is not lost at each step, subdiagonal elmination through Givens to give the decomposition.
• Improvement 2: convergence depends upon . and same eigenvectors and eigenvalues differ by .
• Choose so that small ( close to ).
• not known, good or eigenvalue of lower block of .
• .
• Cubic convergence.
• Implicit lemma: A sym. non-singular , positive off-diagonal elements then and are determined as soon as the first col. of is.
• .
• .
• .
• First row of is that of because only one to touch first row.

Subsections

Next: Example of the Power Up: Principal Components Previous: Principal Components   Index
Susan Holmes 2002-01-12