Can a Matrix Be Expressed as the Sum of a Diagonalizable and a Nilpotent Matrix?

[MathIdiot]

1. write the given matrix A as the sum of a diagonalizable matrix and a nilpotent matrix.

A =

7, 3, 3, 2
0, 1, 2,-4
-8,-4,-5,0
2, 1, 1, 3

Homework Equations

3. The eigenvalues are 1, -1, 3 and the associated eigenvalues are

(1,-2,0,0)
(0,1,-1,0)
(1,-2,0,1), respectively.

Some steps in solving this question/ types of these questions would be amazing. Please explain in layman's terms, because I'm a little over my head :)

 

[Unassuming]

Using your notation,

B=

(0,3,3,2)
(0,0,2,-4)
(0,0,0,0)
(0,0,0,0)

Then B is nilpotent. Likewise for the bottom half. But do they commute?