# Normal modes of oscillator

#### ismaili

1. Homework Statement

I'm reading Landau's Mechanics, in section 23, he discusses the oscillations with more than one degree of freedom, the Lagrangian is

$$L = \frac{1}{2}\left(m_{ik}\dot{x}_i\dot{x}_k - k_{ik}x_ix_k\right)$$

where $$m_{ik},k_{ik}$$ are symmetric constants, and the summation over $$i,k$$ in the above equation is understood.
By substituting the form of solutions

$$x_k = A_k\exp(i\omega t)$$

we get the system of linear equations,

$$\sum_i\left(-\omega^2m_{ik} + k_{ik}\right)A_k = 0\quad\cdots(*)$$

In order to have non-trivial solution, the determinant of the following matrix should be zero,

$$\left| k_{ik} - \omega^2m_{ik} \right| = 0\quad\cdots(**)$$

The roots of $$\omega$$ are denoted as $$\omega_\alpha$$.
Then comes my question. He said that "The frequencies $$\omega_\alpha$$ having been found, we substitute each of them in eq(*) and find the corresponding coefficients $$A_k$$. If all the roots $$\omega_\alpha$$ of the characteristic equation are different, the coefficients $$A_k$$ are proportional to the minors of the determinant (eq(**)) with $$\omega=\omega_\alpha$$."

My question is the sentence with the underline, I think this is a problem of linear algebra actually, but I can't come up with any idea and can't find the material in Wiki.

Thanks for your solution or reference!

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