Kunze Linear Algebra · Problem 2

Elementary Canonical Forms

Mathematics — Methods · diagonalizable-nilpotent-decomposition

Problem

Let

T

be the linear operator on

R^3

which is represented by the matrix

\begin{bmatrix} 3 & 1 & -1 \\ 2 & 2 & -1 \\ 2 & 2 & 0 \end{bmatrix}

in the standard ordered basis. Show that there is a diagonalizable operator

D

R^3

and a nilpotent operator

N

R^3

such that

T = D + N

and

DN = ND

. Find the matrices of

D

and

N

in the standard basis. (Just repeat the proof of Theorem 12 for this special case.)