I think it is well known that the operator
$\frac{EI}{\rho} \frac{\partial^4}{\partial x^4}$ which arises from a standard Euler-Bernoulli beam is self-adjoint in $H$, where $H = L^{2}$, given clamped boundary conditions at one end and no moment or shear force at the other. We define the weighted inner-product on $H$ to be $\langle \cdot ,\cdot \rangle_{S} = \langle \rho \cdot, \cdot\rangle_{L^2} $
Consider the undamped, dynamic beam equation
$\rho \ddot{y}(t,x) + EI \frac{\partial^4y(t,x)}{\partial x^4} = F $
In a FEM simulation we find that $M^{-1}K$ is not symmetric. Here $M$ is the mass matrix and $K$ is the stiffness matrix. We know that both $M$ and $K$ are symmetric by construction with basis functions. Should $M^{-1}K$ be symmetric since the infinite dimensional operator is self-adjoint?