# If a space is the direct sum of invariant subspaces

Put content here**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:
\[[T] = egin{pmatrix} A_1 & 0 & \cdots & 0 \ 0 & A_2 & \cdots & 0 \ dots & dots & \ddots & dots \ 0 & 0 & \cdots & A_k \end{pmatrix}\]
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where each block \(A_i\) represents the restriction \(T|_{W_i}: W_i 	o W_i\).
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**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 \(i 
eq j\)).
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**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.

# Parents

* Eigenvalues and eigenvectors