Expression with Kronecker Delta over upper triangular Matrix

Question

I am having trouble evaluating the following expression involving Kronecker delta's over a double sum restricted over the upper triangular terms: $$\sum_{c_1=0}^N\sum_{c_2\ge c_1}^N \delta_{c_2,l}\delta_{c_2,k}P_{c_1,c_2}$$ I don't think I can naively just equate $c_2=l=k$?

Answer 1

$\delta_{c_2,l}\delta_{c_2,k}$ vanishes except if both Kronecker symbols are equal to $1$. Which is the case if and only if $c_2=k=l$.

Therefore, either $k=l$ and the given sum is equal to

$$\sum_{c_1=0}^k P_{c_1,k}$$ or $k \ne l$ and in that case, the given sum is equal to zero.