qA subspace S Ì Cn is an A-invariant subspace if AxÎ S for every xÎS.
• For example, {0}, Cn , Ker A, and Im
A are all A-invariant subspaces.
• Let l and x be an eigenvalue and a
corresponding eigenvector of AÎCn´n. Then S := span{x} is an A-invariant
subspace since
• Ax=lxÎ S.
• In
general, let {l1, l2, …, lk} (not necessarily distinct) and xi be a set of eigenvalues and a set
of corresponding eigenvectors and the generalized eigenvectors.
Then S=span{x1,…,xk} is an invariant subspace provided that all
the lower rank generalized eigenvectors are included.