# Which space do natural square-symmetric matrices belong to?

Previously, I had asked about Sum preserving transformations of square symmetric matrices with natural elements. Upon further research, I found that the space of square symmetric matrices with natural elements can be generated with simpler matrices $$[a_{i,j}]_{n\times n}$$ such that $$a_{k,l}=a_{l,k}=1$$ keeping other entries zero for $$l,k\leq n$$ with the, only, operations being (1) addition of matrices and (2) multiplication with natural numbers.

As an example, to construct $$2\times2$$ square symmetric matrices with positive elements, I could use the following matrices, with the operators $$+$$ and multiplication with natural numbers, to construct all such matrices: $$\begin{bmatrix} 1&0\\ 0&0\\ \end{bmatrix}, \begin{bmatrix} 0&0\\ 0&1\\ \end{bmatrix}, \text{ and } \begin{bmatrix} 0&1\\ 1&0\\ \end{bmatrix}$$

• Answers to this question concerns with matrices with real, and not natural, elements; therefore, it is not a duplicate.
• The Wikipedia article on Symmetric Matrices does not include these.
• I read about positive-definite matrices, and I don't think this is what I am looking for.

Where could I read about such matrices? What exactly am I looking for?

• Let's call a matrix "elementary symmetric" if it is symmetric and has only one nonzero entry (which must be on the diagonal) or it has exactly two nonzero entries, neither one being on the diagonal; further, insist the nonzero entries all be $1$. then your matrices are the positive linear combinations of the elementary symmetric matrices; you get natural number entries if you restrict to positive integer linear combinations (which are the same as sums). Nov 26, 2023 at 5:54
• @GerryMyerson yes, the 'elementary symmetric' matrices do indeed, as positive linear combinations, generate the matrices of interest to me. However, I am still unclear what these are, what all can I do with these matrices, where to find these, and what algebra is associated with such matrices. In short, what is the entire scope of such matrices and which resources will be helpful in a better understanding of these matrices. Nov 26, 2023 at 7:44
• If you don't know anything you can do with them, why are you interested in them? Nov 26, 2023 at 8:21
• @GerryMyerson, as said in quesion, I am researching sum preserving transformations of such matrices. Further, these are, in a a way, representation of Lewis structures which are fundamental to Chemistry. To say it, we are discussing both rigourous mathematical concepts and applied mathematics. Nov 26, 2023 at 12:09