Suppose that $G$ is a signed graph (a graph in which each edge has a positive of negative sign)
Define Adjacency matrix of $G$ as before except that we have $+1$ for positive edges and $-1$
Prove that the eigen values of Adjacency matrix will not change if you multiply some rows and the corresponding columns by $-1$
Suppose there are $n$ vertices, and let $J\subset \{1,2,\ldots, n\}$ be the indices of the rows/columns we are going to multiply by $-1$. Let $E_J$ be the diagonal matrix such that $e_{ii}=-1$ if $i\in J$, $e_{ii}=1$ if $i\not\in J$, and $e_ij=0$ if $i\neq j$. Then multiplying the rows and columns in $J$ by $-1$ is the same as conjugating by $E_J$. Since conjugation preserves eigenvalues, we are done.