Through chapter 3 of Group Theory by Morton Hamermesh in part 3-6 (Equivalent representations; characters.) I stopped in some point. It's told "If we change the basis in the n-dimensional space $L$, the matrices $D(R)$ will be replaced be their transforms by some matrix $C$. The matrices $$D^{'}(R) = C D(R) C^{-1}$$ also provide a representation of the group $G$, which is equivalent to the representation $D(R)$. What we wish to find are quantities which are intrinsic properties of $D(R)$, i.e., are invariant under change of coordinate axes. One such invariant is easily found, for if we take the sum of the diagonal elements of the matrix, we obtain $$\sum_{i}(CD(R)C^{-1}))_{ii} = \sum_{ikl} C_{ik}D_{kl}(R)C_{li}^{-1 }= \sum_{kl} \delta _{kl}D_{kl}(R)= \sum_{k} D_{kk}(R)$$ Thus the sum of the diagonal elements of matrix $D(R)$ is invariant under a transformation of coordinate axes."
Now, this is a stop point for me. I can't understand how $\delta_{kl}$ appears in last relation. Does matrix $C$ is diagonal?
The equality $C^{-1}C=Id$ gives, by taking the $(l,k)$ component :$$\sum_{i} C^{-1}_{li}C_{ik}=\delta_{lk};$$now multiply this by $D_{kl}$ and sum over $k,l$ and you get the middle equality in your equation.