diagonalisation as well as nilpotent

Question

Confused with this one. Matrix A is not diagonalisable so how can we do this ?

$$LN+N^2=AN$$ $$N(L+N)=AN$$ $$NA=AN$$ similarly, $$LA=AL$$ what to do now ?

Answer 1

Assumption: we are working over the complex numbers.

Let your $A=\begin{bmatrix} X & J\\O & Y\end{bmatrix}$.

Now set $L=\begin{bmatrix} X & O\\O & Y\end{bmatrix}$ and $N=\begin{bmatrix} O & J\\O & O\end{bmatrix}$ so that $A=L+N$.

Now $N^2=O$, and as each of $X$ and $Y$ has characteristic polynomial $\lambda^2+2$ with distinct roots, each is diagonalisable and so therefore is $L$.

It remains to check that $LN=NL$, that is to say that $XJ=JY$. This is a simple calculation.