Invariant
Subspaces

qA subspace *S *Ì **C**n is an *A***-invariant subspace** if *Ax*Î *S* for every *x*Î*S*.

For example, {0}, **C**n , Ker *A*, and Im
*A* are all *A*-invariant subspaces.

Let l and *x* be an eigenvalue and a
corresponding eigenvector of *A*Î**C**n´n. Then *S* := span{*x*} is an A-invariant
subspace since

In
general, let {l*1*,* *l*2*,* **…*,* *l*k*} (not necessarily distinct) and *x**i *be a set of eigenvalues and a set
of corresponding eigenvectors and the generalized eigenvectors.
Then *S*=span{*x**1**,…,x**k*} is an invariant subspace provided that all
the lower rank generalized eigenvectors are included.