Here's a problem I got in my exam , it's a little bit confusing because I had never worked before with polynomials with two variables acting on an endomorphism so I didn't do well in my exam :
Let $k$ be an even integer .Let $E_k$ be the $\mathbb{C}$-the vector space of homogeneous polynomials with two variable and degree $k$ .if $A$ is a matrice from $M_2(\mathbb{C})$ we denot $f_A$ the endomorphisme such that its matrix in the canonical basis is $A$ for $P$ in $E_k$ we denote : $[P!A]=Pof_A$ , thus if $A=\begin{bmatrix}a&b\\c&d\end{bmatrix}.$ we will have $[P!A](X,Y)=P(aX+bY,cX+dY)$
let $S=\begin{bmatrix}0&1\\-1&0\end{bmatrix}.$ and $T=\begin{bmatrix}1&1\\0&1\end{bmatrix}.$
$A_k=\{P\in E_k ; [P!S]+P=0\}$
$B_k=\{P\in E_k ,[P!STST]+[P!ST]+P=0\}$
1) construct a basis of $A_k$ and calculate its dimension .
2)diagonalise the matrix $ST$ (easy)
3)Deduce the diagonalisation of the endomorphism $P\to [P!ST]$ of $E_k$
4)construct a basis of $B_k$ and calculate its dimension