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##

Eigendecompositions

Define
and
. is called an *eigenvector matrix* of .
Since the are orthogonal unit vectors, we see that ; i.e.,
is a *unitary (orthogonal) matrix*.
The equalities
for may also be written
or
. The factorization

is called the *eigendecomposition* of . In other words,
is similar to the diagonal matrix , with similarity
transformation .
If we take a subset of columns of (say
=
columns 2, 3, and 5), then these columns span an invariant subspace of .
If we take the corresponding submatrix
of , then we can write the corresponding
*partial eigendecomposition* as
or
. If the columns in are replaced by
different vectors spanning the same invariant subspace, then we get
a different partial eigendecomposition
,
where is
a -by- matrix whose eigenvalues are those of , though
may not be diagonal.

** Next:** Conditioning
** Up:** Hermitian Eigenproblems J.
** Previous:** Equivalences (Similarities)
** Contents**
** Index**
Susan Blackford
2000-11-20