We discuss four eigendecompositions, or four pencils that are equivalent
to and for which it is simpler to solve eigenproblems.
This section is analogous to §2.5.4.
The first eigendecomposition, diagonal form,
exists only when there are independent eigenvectors.
The second one, Weierstrass form, generalizes diagonal form to all
pencils where the characteristic polynomial
is not identically 0.
We can also describe the Weierstrass form as the generalization of
the Jordan form matrices to pencils.
The Weierstrass form, like the Jordan form,
can be very ill-conditioned (indeed, it can change discontinuously),
so for numerical purposes we typically use the third one,
generalized Schur form,
which is cheaper and more stable to compute. We briefly mention a fourth
one, which we call the Weierstrass-Schur form,
that is as stable as the Schur form but computes some of
the detailed information about deflating subspaces provided by the Weierstrass
form.
Define
.
If there are
independent right eigenvectors
,
we define
.
is called a (right) eigenvector matrix of
.
Similarly let
be a left eigenvector matrix.
The
equalities
and
for
may also be written
and
.
and
may furthermore be chosen so that
and
are both diagonal, and
.
The factorization
If we take a subset of columns of
and of
(say
= columns 2, 3, and 5
and
)
then these columns span a pair of deflating subspaces of
.
If we take the corresponding submatrices
and
,
then we can write the corresponding
partial diagonal form as
and
.
If the columns in
and
are replaced by
different vectors spanning the same deflating subspaces, then we get
a different partial eigendecomposition
and
,
where
is
a
by
pencil whose eigenvalues are those of
,
though
may not be diagonal.
Similar procedures for producing partial eigendecompositions work
for the other eigendecompositions discussed below.
If all the are distinct, then there are
independent
eigenvectors,
and the diagonal form exists. This is the simplest and most common case.
For example, if one picks
and
``at random,''
the probability is 1 that the eigenvalues are distinct.
A diagonal form of the pencil
in (2.3)
in §2.5.4
does not exist,
since it has just one independent eigenvector. Instead, we can compute its
Weierstrass form,
which is a decomposition
Unfortunately, the Weierstrass form is generally not suitable for numerical computation, for the same reason that the Jordan form is not suitable. See §2.5.4 for discussion.
So instead we use eigendecompositions of the form
Finally, we consider the Weierstrass-Schur form. It is quite complicated, so we only summarize its properties here. Like the Schur form, it only uses unitary (orthogonal) transformation and so can be computed stably. Like the Weierstrass form, it explicitly shows what the sizes of the (Jordan) blocks are and gives explicit bases for many more invariant subspaces than the Schur form.
Many textbooks give explicit solutions for problems such as solving
ordinary differential equations
in terms of the Weierstrass form of
[114].
These methods are
to be avoided numerically, because of the difficulty of computing the
Weierstrass form.
Nearly all these problems have alternative solutions in terms of the
generalized Schur form,
or in some cases the Weierstrass-Schur form.