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.
If there are independent right eigenvectors
is called an eigenvector matrix of .
for may also be written
. 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 . 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.