Eigendecompositions

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

is called the

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

where is a block-diagonal pencil, with one or more upper triangular diagonal blocks for each eigenvalue. When there are no infinite eigenvalues, this is identical to the Jordan form discussed in §2.5.4. When there are infinite eigenvalues, there are blocks quite similar to Jordan blocks with a single infinite eigenvalue and one right and left eigenvector.

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

where and are unitary (or orthogonal) matrices and is triangular (or quasi-triangular). This is called the

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.