= Solution
The four spring extensions are $x_1$, $x_2-x_1$, $x_3-x_2$ and $-x_3$. <Hooke's law> therefore gives <potential energy> $k[x_1^2+(x_2-x_1)^2+(x_3-x_2)^2+x_3^2]/2$, while the <kinetic energy> is $m\sum_i\dot x_i^2/2$. Thus
$$
\boxed{L=\frac m2\sum_{i=1}^3\dot x_i^2-\frac k2[x_1^2+(x_2-x_1)^2+(x_3-x_2)^2+x_3^2],}
$$
with the matrices in the requested <quadratic form>
$$
\boxed{T=mI_3,\qquad V=k\begin{pmatrix}2&-1&0\\-1&2&-1\\0&-1&2\end{pmatrix}.}
$$
The equilibrium offsets disappear because the expansion is in displacements from static equilibrium.
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