If $L$ is a nonsymmetric Laplacian matrix, is it possible that $L+L^T$ is a positive definite matrix?
2026-03-25 00:01:11.1774396871
Is it possible that $L+L^T$ is a positive definite matrix?
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The answer is no. It is impossible for $L+L^T$ to be a positive definite matrix. This is because of \begin{align} 1^T(L+L^T)1 &= 1^TL1 + 1^T L^T 1 \\ &= 1^T(L1) + (L1)^T 1 \\ &= 0 \end{align} which has used the property $L1=0$. Therefore, this contradicts the definition of a positive definite matrix. This means $L+L^T$ has at least one non-positive eigenvalue.