What is the physical meaning of the eigenvectors of the mass matrix?
If I consider a 2-dof system with one mass linked to two orthogonal springs and I write the equations in any orthogonal system of coordinates, and get a mass matrix $M$. The eigenvectors correspond to the direction of the springs and the eigenvalues of $M$ correspond to the mass. If I am not too stupid and I write the equations "naturally", I directly get a diagonal matrix $$\begin{bmatrix} m & 0 \\ 0 & m \end{bmatrix}$$
So in this simple case, the physical interpretation of the eigenvalues and eigenvectors of the mass matrix is obvious.
Now, instead of considering ponctual masses, I choose a mass density (for e.g. using FEM: the component $(i,j)$ will be given by $m_{ij}=\int \bar{m}\varphi_i(x)\varphi_j(x)\text{d}x$ with $\varphi$ basis functions). The mass matrix will be symmetric positive definite but no longer diagonal. But what is the physical meaning of its eigenvalues and eigenvectors in this case?
Note that I am asking this in the context of Newtonian mechanics, but it seems to be mostly used in other fields: cf google. Hopefully we will get several answers from several fields.
Example of construction of a non-diagonal mass matrix, as in finite-element method.
Consider two bar elements with mass density $\bar m$ and unit length. I assume that for each element, the displacement between the two nodes is linear, which gives to basis function: $$ \varphi_1(x)=1-x \quad \varphi_2(x)=x $$
Then, it can be proven (using the principle of virtual work) that the mass matrix of each element is given by M_e, of entries $m_{ij}=\int_0^1 \varphi_i(x)\varphi_j(x)\text{d}x$. The calculation yields:
$$M_e=\left( \begin{array}{cc} \frac{1}{3} & \frac{1}{6} \\ \frac{1}{6} & \frac{1}{3} \\ \end{array} \right)$$
Altogother, for my two elements, the mass matrix is: $$M=\left( \begin{array}{ccc} \frac{2}{3} & \frac{1}{6} & 0 \\ \frac{1}{6} & \frac{2}{3} & \frac{1}{6} \\ 0 & \frac{1}{6} & \frac{1}{3} \\ \end{array} \right) $$
The eigenvalues of $M$ are $ 0.861681, 0.551851, 0.253134 $ and the eigenvectors are $$(0.631781,0.739239,0.233192),(0.755789,-0.520657,-0.397113),(0.172148,-0.427132,0.88765)$$ What is their physical meaning?