Diagonalizable Part of Triangular Matrix

Definition

• Suppose $T$ is Triangularizable
• Let $E_i$ be the Projection onto $W_j=ker(p_j(T{)}^{r_j})=range(E_j)$
• Then, $D=c_1{E}_1+\cdots +c_k{E}_k$ is a Diagonal Matrix, and the diagonalizable part of $T$

Example

2 & 1 \\ 0 & 2 \\ \end{array}\right]$$ Diagonal part is

D = \left[\begin{array}{cc} 2 & 0\ 0 & 2\ \end{array}\right]