We discuss four eigendecompositions, or four matrices that are similar to
and for which it is simpler to solve eigenproblems.
The first one, diagonal form,
exists only when there are
independent eigenvectors.
The second one, Jordan form, generalizes diagonal form to all
matrices. It can be very ill-conditioned (indeed, it can change discontinuously),
so for numerical purposes we typically use the third one, Schur form,
which is cheaper and more stable to compute. We briefly mention a fourth
one, which we call Jordan-Schur form,
that is as stable as the Schur form but computes some of
the detailed information about invariant subspaces provided by the Jordan form.
Define
.
If there are
independent right eigenvectors
,
we define
.
is called an eigenvector matrix of
.
The
equalities
for
may also be written
or
. The factorization
If we take a subset of columns of
(say
=
columns 2, 3, and 5), these columns span an invariant subspace of
.
If we take the corresponding submatrix
of
, and the corresponding three rows of
(say
),
we can write the corresponding
partial diagonal form as
or
. If the columns in
are replaced by
different vectors spanning the same invariant subspace, we get
a different partial eigendecomposition
,
where
is
a
by
matrix whose eigenvalues are those of
, though
may not be diagonal. Similar procedures for producing partial
eigendecompositions for the other eigendecompositions discussed below.
If all the are distinct, there are
independent eigenvectors,
and the diagonal form
exists. This is the simplest and most common case.
For example, if one picks a matrix ``at random,''
the probability is 1 that the eigenvalues are distinct.
A diagonal form of the matrix in (2.3) does not exist,
since it has just one independent eigenvector. Instead, we can compute its
Jordan form, which is a decomposition
Unfortunately, the Jordan form is generally not suitable for numerical
computation. Here are two reasons. First, it can change discontinuously
as changes. For example, the matrix
for
with
is
, but
for
,
itself. Second, the eigenvector matrix
can be very ill-conditioned, i.e., hard to invert accurately.
In the case of
, the condition number of
grows like
.
So instead we use eigendecompositions
of the form
Finally, we consider the Jordan-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 Jordan form, it explicitly shows what the sizes of the Jordan blocks are and gives explicit bases for many more invariant subspaces than Schur form.
Many textbooks give explicit solutions for problems such as computing the matrix
of the exponential in terms of the Jordan form of
[114].
These methods are
to be avoided numerically, because of the difficulty of computing the
Jordan form.
Nearly all these problems have alternative solutions in terms of the Schur form,
or in some cases the Jordan-Schur form.