Linear Algebra, 2/e, 2nd Edition

5.5  INVARIANT SUBSPACES

A deeper analysis of linear operators on a vector space depends on subspaces on which they act as linear operators again. The eigenspaces of a linear operator are examples of such spaces. If W is an eigenspace of a linear operator T belonging to an eigenvalue λ, then for any w ∈ W, the image Tw is again a vector in W, for T(Tw) = T(λw) = λTw. Thus, T maps W into W, and we express this property of W with respect to T by saying that W is invariant under T.

Definition 5.5.1.    Let T be a linear operator on a vector space V. A subspace W of V is said to be T-invariant if for any w ∈ W, Tw ∈ W. In other words, W is T-invariant if T(W) ⊂ W.

This definition is valid for infinite-dimensional vector spaces too. It is clear that ...