I have the linear transformation $L_1: V \rightarrow V$ (where $V = \mathbb{C}[x,y]$ is the vector space of polynomials with complex coefficients) defined as:
$L_1(f(x,y)) = \frac{f(x,y)+f(y,x)}{2}$
I want to find out if $L_1$ is diagonalizable. To do this I first tried to find the matrix representation of $L_1$ by constructing a standard basis $\{1,x,y,x^2,xy,y^2,...\}$ for $V$ and then applying the transformation on this basis. However, this approach feels very clumsy. Is there any other way I could approach this problem?
Thanks!