A symmetric matrix is a way to describe a weighted, undirected graph: each edge in the graph is assigned a weight [latex]W_{ij}[/latex]. Since the graph is undirected, the edge weight is independent of the direction (from [latex]i[/latex] to [latex]j[/latex] or vice-versa). Hence, [latex]W[/latex] is symmetric.
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To the graph in the figure, we can associate the corresponding undirected graph, obtained by ignoring the direction of the arrows. Assuming that all the edges have the same weight, the undirected graph has the edge weight matrix given by [latex]W=\left[\begin{array}{cccccccc} 1 & 1 & 0 & 0 & 0 & 0 & 0 & -1 \\ -1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 \\ 0 & -1 & -1 & -1 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 & 1 & 0 \\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 \end{array}\right][/latex] |
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