Three particles of unit mass move along a line in a potential

$$V=\frac{1}{2}\left(\left(x_{1}-x_{2}\right)^{2}+\left(x_{1}-x_{3}\right)^{2}+\left(x_{3}-x_{2}\right)^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\right)$$

where $x_{i}$ is the coordinate of the $i$ th particle, $i=1,2,3$.

Write the Lagrangian in the form

$$\mathcal{L}=\frac{1}{2} T_{i j} \dot{x}_{i} \dot{x}_{j}-\frac{1}{2} V_{i j} x_{i} x_{j}$$

and specify the matrices $T_{i j}$ and $V_{i j}$.

Find the normal frequencies and normal modes for this system.