Notation for summing all elements under the diagonal of a square matrix

Question

I have a simple question:

What is the notation for summing all elements under the diagonal of a square matrix?

I appreciate your help.

Answer 1

The sum of the diagonal element of a square matrix is its trace. There are many notations in use for the trace of a matrix $M$, but the ones that I see most frequently are $\text{Tr}(M)$, $\text{tr}(M)$, and sometimes $\text{Sp}(M)$, which is a reference to the German word Spur for the same concept. Some will also just write out $\text{Trace}(M)$ or $\text{trace}(M)$, but this is less common in my experience.

Edit: As Christopher Ernst let me know, I may have misunderstood your question. It sounds like you're asking for a notation/word for the sum of the elements of the subdiagonal of a matrix, and unfortunately I don't believe I have seen a term for it in the literature (or at least one that is widely in use). Hopefully someone can come along with a canonical name/notation for it.

Answer 2

If $M=\left[M_{ij}\right]$ is an $n\times n$ matrix, one way to notate the sum of all entries under the diagonal is

$\displaystyle \sum_{1\le j<i\le n} M_{ij}$

Another way:

$\displaystyle \sum_{i=2}^n\sum_{j=1}^{i-1} M_{ij}$