Mass-weighted orthogonality of normal modes ($u_i^{T}M{u}_j=0({\omega }_i^2\ne {\omega }_j^2)$)

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 $K{u}_i={\omega }_i^{2}M{u}_i$. Taking the scalar product with $u_j$ and using symmetry in the two equations gives $({\omega }_i^2-{\omega }_j^2)u_i^{T}M{u}_j=0$. 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.