Let $G = -D^{-1}(L+L^T)$ where $D$ is a diagonal matrix, and $L$ is a lower triangular matrix with $0$ on the diagonal. Find a symmetric matrix similar to G (so find a matrix $S$ such that $G = P^{-1}SP$ where $P$ is any invertible matrix). Assume all matrices are square.
Solution: Let $P = D^{\frac{1}{2}}$ and $S = PGP^{-1}.$ Clearly since $$S = PGP^{-1} = -D^{-\frac{1}{2}}(L+L^T)D^{-\frac{1}{2}}$$ is also symmetric we have our answer.