Invariant Direct Sum of Linear Transformations Theorem

Theorem

1. Let $T\in \mathcal{L}(V)$
2. Suppose $V=W_1\oplus \cdots \oplus W_k$
3. Let $E_1,\dots ,E_k$ be Projections with $range(E_j)=W_j$
4. Then each $W_j$ is invariant for $T⟺T{E}_j=E_{j}T,\forall 1\le j\le k$

Proof

1. Suppose $T$ commutes with each $E_i$
2. Then, by Block Diagonal Basis for Direct Sum of Linear Transformations theorem, as $W_i=range(E_i),W_i$ is $T$ Invariant
3. Suppose $W_i$ is $T$ Invariant for all $i$. Since $E_i\alpha \in W_i$ and $W$ is invariant under $T$
4. Then, $T{E}_i\alpha =P_i\in W_i$
5. Now, consider $E_{i}T\alpha$. By construction, $\alpha =E_1\alpha +\cdots +E_k\alpha$
6. Then, $T\alpha =T{E}_1\alpha +\cdots +T{E}_k\alpha$
7. As $T{E}_j\alpha ={\beta }_j\in W_j,\forall j$
8. Then,

$$E_{i}T{E}_j\alpha ==\{\begin{array}{ll}{\beta }_j& i=j\\ 0& i\ne j\end{array}$$

9. $E_{i}T\alpha =E_i(T{E}_i\alpha )+\cdots +E_i(E_k\alpha )=0+\cdots +{\beta }_j+\cdots +0$

Intuition

We are $T$ invariant $⟺$ there is a series of projections in the whole space that commute