Linear Algebra, 5th Edition

5.4 Invariant Subspaces and the Cayley-Hamilton Theorem

In Section 5.1, we observed that if v is an eigenvector of a linear operator T, then T maps the span of $\left\{v\right\}$ into itself. Subspaces that are mapped into themselves are of great importance in the study of linear operators (see, e.g., Exercises 29-33 of Section 2.1).

Definition.

Let T be a linear operator on a vector space V. A subspace W of V is called a T-invariant subspace of V if $\text{T}(\text{W})\subseteq \text{W}$, that is, if $\text{T}(v)\in \text{W}$ for all $v\in \text{W}$.

Example 1

Suppose that T is a linear operator on a vector space V. Then the following subspaces of V are T-invariant:

1. (1) $\left\{0\right\}$

2. (2) V

3. (3) R(T)

4. (4) N(T)

5. (5) ${\text{E}}_{\lambda }$, for any eigenvalue $\lambda$ of T.

The proofs that these subspaces are T-invariant are left as exercises. (see Exercise ...