Direct Sum of Linear Transformations

Definition

• Suppose $T\in \mathcal{L}(V)$
• Suppose $V=W_1\oplus ..\oplus W_k$ where each $W_i$ is Invariant
• let $T_j$ be the restriction of $T$ to $W_j$
• As $V$ is a direct sum, then every vector $\alpha \in V$ is represented uniquely by vectors in $W_1,\dots ,W_k$. That is to say, $\forall \alpha ,{\alpha }_1+\dots {\alpha }_k\in V$
• Applying to $T$ gives: $T(\alpha )=T({\alpha }_1+\cdots +{\alpha }_k)=T({\alpha }_1)+\cdots +T_k({\alpha }_k)$
• Then, we say that $T$ is the direct sum of $T_j\in L(W_j)$ and write $T=T_1\oplus \cdots \oplus T_k$

Theorems

• Block Diagonal Basis for Direct Sum of Linear Transformations
• Invariant Direct Sum of Linear Transformations Theorem