Symmetric Real Matrices
Definition
A square real matrix is symmetric (we call it a symmetric matrix) if its values do not depend on the order of the indices. In other words, a matrix is symmetric if the values above and below the diagonal are a mirror image.
Examples
The matrix
\[
\bmat{1 & 2 \\ 2 & 3}
\] \[
\bmat{1 & 2 \\ 3 & 4}
\]
Notation
Suppose $A \in \R ^{n \times n}$ with $A^\top
= A$ (i.e., symmetric).
We denote the set of all such symmetric $n
\times n$ real matrices by $\mathbfsf{S} ^n$.
So
\[
\mathbfsf{S} ^n = \Set*{A \in \R ^{n \times n}}{A^\top = A}.
\]