However, the
subspaces *S**2*=span{*x**2*}, *S**23*=span{*x**2
**, x**3*}, *S**24*=span{*x**2** , x**4*}, *S**234*=span{*x**2** , x**3** , x**4*} are not *A*-invariant subspaces since the lower rank
generalized eigenvector *x**1* of *x**2* is not in these subspaces.

To
illustrate, consider the subspace *S**23*. It is an *A*-invariant subspace if *Ax**2** *Î *S**23*. Since *Ax**2*=l*1**x**2** *+*x**1* , *Ax**2** *Î *S**23* would
require that *x**1 *be a linear combination of *x**2* and *x**3*, but this is impossible since *x**1* is independent of *x**2* and *x**3*.