Let $T$ be a tensor of type $(2,2)$ which is invariant under arbitrary transformation. Find the general form of $T$.
To be honest I do not know how to solve this problem at all. But I can say for sure the following: since $T\in T_2^2(V)$ then for any matrix $C=(c_i^j)$ with $\det C\neq 0$ we have $$T_{\alpha,\beta}^{\gamma,\delta}=c_{\alpha}^i c_{\beta}^{j}d_k^{\gamma} d_l^{\delta} T_{k,l}^{i,j},$$ where $d^j_i$ are elements of matrix $D=C^{-1}$.
And probably I have to play with different matrices $C$ and find out the general form of our tensor $T$.
Can anyone show the full solution, please? I will appreciate your help and answer!