= Central spring stiffness matrix
{title2=$V=\tfrac12q^TKq,\quad K_{ij}=-k_{ij}n_{ij}n_{ij}^T\ (i\ne j)$}
For point masses joined by unstretched central <springs> obeying <Hooke's law>, let $n_{ij}$ be the equilibrium <unit vector> along an edge and $u_i$ the small <displacement> of its endpoint. The leading extension is $n_{ij}\cdot(u_j-u_i)$, so its quadratic <elastic potential energy> is $k_{ij}[n_{ij}\cdot(u_j-u_i)]^2/2$. Consequently the block <stiffness matrix> has off-diagonal blocks $K_{ij}=-k_{ij}n_{ij}n_{ij}^T$ and diagonal blocks $K_{ii}=\sum_jk_{ij}n_{ij}n_{ij}^T$, with absent edges contributing zero. It is <symmetric> and <positive semidefinite>. The <normal modes> solve $Kq=\omega^2Mq$, where $M$ is the <mass matrix>. Rigid translations and infinitesimal rotations are <rigid-motion zero modes of a spring framework>. A prestressed spring can add transverse quadratic terms, so the unstretched assumption matters.
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