Prove Laplacian matrix is positive semidefinite and has an eigenvector of $[1,1,...,1]$

Question

Show that for a symmetric matrix $W \in R^{n \ x \ n}$ and a diagonal matrix $D$ defined by $D_{ii}= \sum_{j=1}^n w_ij$, if $W_{ij} \ge 0$ for $\forall \ i \ge 1$ and $j \le n$, then $L = D-W$ (called the Laplacian matrix) is positive semidefinite, and has an eigenvector of $[1,1,...,1]$

Answer 1

Check that $L\mathbf{1}$ is coming out to be equal to $0\cdot\mathbf{1}$, because each row sum is zero here.