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

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SUMMARY

The matrix A can be expressed as the sum of a diagonalizable matrix and a nilpotent matrix. The given matrix A is defined as follows: A = [[7, 3, 3, 2], [0, 1, 2, -4], [-8, -4, -5, 0], [2, 1, 1, 3]]. The eigenvalues of A are 1, -1, and 3, with corresponding eigenvectors that help in identifying the diagonalizable component. The nilpotent matrix B is defined as B = [[0, 3, 3, 2], [0, 0, 2, -4], [0, 0, 0, 0], [0, 0, 0, 0]], which satisfies the nilpotency condition.

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MathIdiot
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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 :)
 
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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?
 

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