Basis to Generate Invariant Subspace Diagonal Block

Theorem

• Let $T\in \mathcal{L}(V)$
• Let $c_1,\dots ,c_n$ be distinct Eigenvalues of $T$
• Then, there exists a basis $\beta$ such that:

[T]{\beta} = \left[\begin{array}{c|c} diag(c{1}I_{d_{1}},\dots,c_{n}I_{d_{n}}) & C\ \hline 0 & D \end{array}\right]