Invariant Subspaces Give Block Diagonal Matrix Representations

Theorem

1. Let $W$ be Invariant Subspace under Linear Operator $T\in \mathcal{L}(V)$
2. Then, there exists a basis $\beta$ s.t

$$[T{]}_{\beta }=\left[\begin{array}{cc}B& C\\ 0& D\end{array}\right]$$

Corrolaries

• Characteristic and Minimal Polynomial of a Restriction Operator divides those of the Parent Operator
• Basis to Generate Invariant Subspace Diagonal Block