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
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.