= Mass-weighted orthogonality of normal modes
{title2=$u_i^T M u_j=0\quad(\omega_i^2\ne\omega_j^2)$}
= Mass-weighted orthogonality
{synonym}
For a finite conservative linear mechanical system $M\ddot q+Kq=0$ with real symmetric positive definite <mass matrix> $M$ and real symmetric stiffness matrix $K$, the <normal modes> satisfy $Ku_i=\omega_i^2Mu_i$. Taking the scalar product with $u_j$ and using symmetry in the two equations gives $(\omega_i^2-\omega_j^2)u_i^TMu_j=0$. Distinct-frequency modes are therefore orthogonal in the <inner product> defined by $M$, even if the dynamical matrix $M^{-1}K$ is not symmetric in the ordinary Euclidean <inner product>. Within a degenerate mode space one can choose an $M$-orthogonal basis.
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