Why is it true that "$\sum_{k,l\geq1}a_{kl}=\sum_{m\geq1}\sum_{k+l=m}a_{kl}$"?

Question

Why is it true that "$\sum_{k,l\geq1}a_{kl}=\sum_{m\geq1}\sum_{k+l=m}a_{kl}$"? I encounter this identity quite often, but I don't find it very intuitive. Can someone elaborate on this? Also, is convergence of both sides equivalent? Thanks in advance!

Answer 1

$\sum_{k, l \ge 1} a_{kl}$ is summing the elements in a $K \times L$ matrix one by one, assuming $1 \le k \le K$ and $1 \le l \le L$.

$\sum_{k\ge 1} \sum_{l \ge 1} a_{kl}$ is summing by rows - ie sum all the elements in rows $k=1, k=2, k=3, .. k=K$ in turn, then add all $K$ row-sums together.

$\sum_{l\ge 1} \sum_{k \ge 1} a_{kl}$ is summing by columns.

$\sum_{m \ge 1} \sum_{k+l=m} a_{kl}$ is summing the same elements by anti-diagonals. Each of the diagonals is defined by $m$ and the terms in each diagonal are summed before moving to the next diagonal.

Whichever order you add the elements, the sum is always the same.