We generalize real symmetric matrices to arbitray sets.
A square matrix is symmetric (a symmetric matrix) if its values do not depend on the order of the indices.
Suppose $S$ is a nonempty set and $A \in S^{n
\times n}$.
Then $A$ is symmetric means
\[
A_{ij} = A_{ji} \quad \text{for all } i, j \in \set{1,
\dots , n}
\]