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##

Eigendecompositions

Define
and
. is called an
*eigenvector matrix* of .
Since the are unit vectors orthogonal with respect to the inner product
induced by , we see that
, a nonsingular diagonal matrix.
The equalities
for may also be
written
or
.
Thus
is diagonal too.
The factorizations

(or
and
)
are called an *eigendecomposition* of .
In other words, is congruent to the diagonal pencil
, with congruence transformation .
If we take a subset of columns of (say
=
columns 2, 3, and 5), then these columns span an
eigenspace of . If we take the corresponding submatrix
of , and similarly define
,
then we can write the corresponding
*partial eigendecomposition* as
and
.
If the columns in are replaced by
different vectors spanning the same eigensubspace, then we get
a different partial eigendecomposition, where
and
are replaced by different -by- matrices
and
such that the eigenvalues
of the pencil
are
those of
, though
the pencil
may not be diagonal.

** Next:** Conditioning
** Up:** Generalized Hermitian Eigenproblems
** Previous:** Equivalences (Congruences)
** Contents**
** Index**
Susan Blackford
2000-11-20