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
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.