Advantages (from a mathematical perspective) of representing data as symmetric matrices

Question

From Wikipedia, a symmetric matrix is a square matrix that is equal to its transpose. An example of this (I think) is an adjacency matrix with undirected edges, which is a square matrix representing links between elements (undirected in this case).

The thing is, the data is duplicated in this case (looking at it from a computer science perspective). The two halves of the diagonal carry the same information, so you can just use one half of it.

I am new to matrices, so I apologize if this is a basic question. The question is if there is any advantage mathematically or from a computer science perspective to representing data (such as graphs, but more generically any data such as key/value pairs or pointers, etc.) using symmetric matrices like this. Wondering if it (for example) makes lookup O(1) in certain cases, or it provides some other significant hidden benefit because of the symmetry.