Summation of indicator function

Question

I need to calculate this summation. I have tried to solve it myself but can't seem to get anywhere.

I know that the answer needs to be $2q+1-h$.

$$\sum_{j, k=-q}^q 1_{(h+j-k=0)}$$

Answer 1

Consider the matrix $A_{kj}$, $1\leq i,j \leq 2q+1$ with entries $$ A_{kj}=1_{h}(k-j). $$ Then your sum is precisely $\sum_{i,j}A_{ij}$. When $h=0$, only the diagonal is non-zero, so the sum is $2q+1$. When $h=|1|$, you get ones above the diagonal, so the sum is $2q$. When $|h|\geq 2q+1$, the sum is $0$. Can you fill in the remaining cases?