For an complex matrix , does not necessarily have a basis consisting of eigenvectors of . But it will always have a basis consisting of generalized eigenvectors of .

In this section we assume all vector spaces and matrices are complex. As we have already seen, the matrix is not diagonalizable. In fact, , indicating that the single eigenvalue has algebraic multiplicity , while implying that is defective.

However this is not the end of the story. Letting , we see that is the zero matrix. Moreover, , where . We then see that is not an eigenvector of , but is. There is an inclusion In this example, the vector is referred to as a generalized eigenvector of the matrix ; it satisfies the property that the vector itself is not necessarily an eigenvector of , but is for some . One other observation worth noting: in this example, the smallest exponent of satisfying the property is , the algebraic multiplicity of the eigenvalue . This is not an accident.

Letting , we have a sequence of inclusions

The generalized eigenvalue problem is to find a basis for each generalized eigenspace compatible with this filtration. This means that for each , the vectors of lying in is a basis for that subspace.

This turns out to be more involved than the earlier problem of finding a basis for , and an algorithm for finding such a basis will be deferred until Module IV.

One thing that can often be done, however, is to find a Jordan chain. We will first need to define some terminology.

By the above Theorem, such an always exists. A generalized eigenvector of , then, is an eigenvector of iff its rank equals . For an eigenvalue of , we will abbreviate as .

By definition of rank, it is easy to see that every vector in a Jordan chain must be non-zero. In fact, more is true

In this way, a rank generalized eigenvector of (corresponding to the eigenvalue ) will generate an -dimensional subspace of the generalized eigenspace with basis given by the Jordan chain associated with .

The previous examples were designed to be able to easily find a Jordan chain. The following is a bit more involved.