Invariant Subspace Check using Projections

Theorem

• Let $T\in \mathcal{L}(V)$
• Suppose $V=W_1\oplus \cdots \oplus W_k$
• Let $E_1,\dots ,E_k$ be Projections with $range(E_j)=W_j$
• Then, $W_j$ invariant under $T,\forall j\in [1,k]$ $⟺T{E}_j=E_{j}T,\forall j\in [1,k]$