If a space is the direct sum of invariant subspaces

Theorem: If (V = W_1 \oplus W_2 \oplus \cdots \oplus W_k) is a direct sum of (T)-invariant subspaces, then the matrix of (T) (with respect to a basis adapted to this decomposition) is block diagonal:
[[T] = egin{pmatrix} A_1 & 0 & \cdots & 0 \ 0 & A_2 & \cdots & 0 \ dots & dots & \ddots & dots \ 0 & 0 & \cdots & A_k \end{pmatrix}]

where each block (A_i) represents the restriction (T|_{W_i}: W_i o W_i).

Intuition: When a space decomposes into invariant pieces, the transformation acts independently on each piece. The behavior of (T) on (W_i) does not affect vectors in (W_j) (for (i
eq j)).

Example: If (V = E_{\lambda_1} \oplus E_{\lambda_2}) where (E_{\lambda_i}) are eigenspaces, then (T) is represented by a diagonal matrix with (\lambda_1) and (\lambda_2) on the diagonal. This is the essence of diagonalization.