Finding the components of the tensor for potential and kinetic energy

I believe this is the "missing link", stated in a less abstract fashion than in the above comment: https://en.wikipedia.org/wiki/Quadratic_form . Some programs in physics cover that in undergraduate algebra courses, some leave it for later. Notably, this method doesn't apply just to tensors, it's a general connection between symmetric matrices (of spaces $mathcal{M}_{ntimes n}({ℝ})$, since they are quadratic) and equations.

If you have an expression like $Qleft({xi }_{1},dots ,{xi }_{n}right)=sum _{{i}_{i}j=1}^{n}{alpha }_{ij}{xi }_{i}{xi }_{j}$ , which you do since the lagrangian of small oscillations has quadratic terms of both $x$ and $stackrel{.}{x}$, you may transform the quadratic form of the potential energy (your first expression) into a symmetric matrix by putting values of ${alpha }_{ii}$ on the diagonal (in this case the coefficients are ${alpha }_{11}=k$, ${alpha }_{22}=2k$, and ${alpha }_{33}=k$, and, because the coefficients are symmetric (see definition), the matrix is symmetric as well. Hence, other elements are put in their appropriate position ( in this case ${alpha }_{12}={alpha }_{21}=-k$, ${alpha }_{23}={alpha }_{32}=-k$, ${alpha }_{13}={alpha }_{31}=0$). Note also that the matrix coefficients of nondiagonal elements are half of the ${alpha }_{ij}$ coefficients from the quadratic form, since they are represented twice in the matrix.

Hope this helps, if needed I will give some more examples.