Small oscillations

Let

.

Then solutions of the form q_j = Re(A_je^{iw t}) can be found.  We can find the w² from det(k_ij-w ²T_ij) = 0.  For a system with n degrees of freedom, n characteristic frequencies w_a can be found.  Some frequencies may be degenerate.
For a particular frequency w_a we solve

to find the A_ja . 

[While the secular equation det(k_ij-w ²T_ij) = 0 can in principle always be solved, it is often simpler to find the normal modes by using physical insight and noting the symmetries of the system.]

The most general solution for each coordinate q_j is a sum of simple harmonic oscillations in all of the frequencies w_a .

.