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