Every Triangular Matrix is The Sum of The Diagonal and Nilpotent

Theorem

• With $T\in \mathcal{L}(V)$
• With $dim(V)<\infty$
• With the Diagonalizable Part of Triangular Matrix $D\in \mathcal{L}(V)$
• With the Nilpotent $N\in \mathcal{L}(V)$
• Then, $T=D+N$