If we have a set of N-dimensional vectors $A_i$, and a Lagrangian with a set of NxN square matrices $M_{ij}\neq 0$:
$$L = \sum_{ij} A_i M_{ij} A_j\tag{1}$$
That couples these vectors, can this produce equations of motion in which the vector components are uncoupled?
Consider the following, the Lagrangian produces the equations of motion:
$$\sum_n \left( M_{in} + M^T_{ni} \right)A_n=0\tag{2}$$
Now I would really like to put these together:
$$\sum_{n,m} A^T_m\left( M_{jm} + M^T_{mj} \right)\left( M_{in} + M^T_{ni} \right)A_n=0\tag{3}$$
And aks if it is possible to get an equation where the vectors decouple:
$$\sum_n \alpha_{ijn}A^T_nA_n=0\tag{4}$$
Where the $\alpha_{ijn}$ are simply coëfficients the depend on the vector inner-products and on which equations we have combined.
If this is impossible it is perhaps useful to consider complex numbers, although I'm not sure if this makes any difference, I believe we could already have considered real matrices of the kind: $\begin{bmatrix} 0 & -1\\ 1 & 0 \end{bmatrix}$ which can do everything that complex numbers can. Likewise a real number-representation probably exists for quaternions (gamma matrices?) and so forth, so that no real extra possibilities are added by considering these.
Anyway, for the sake of making an effort, for complex vectors:
$$L = \sum_{ij} A^\dagger_i M_{ij} A_j\tag{5}$$
We get equations of motion:
$$\sum_n A^\dagger_n M_{ni}=0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \sum_n M_{in} A_n=0\tag{6}$$
We may or may not like to demand that these are the same equations and that the Lagrangian be real via the requirement $M^\dagger_{ij}=M_{ji}$. I don't really mind as long as the equations can decouple.
This gives the equations:
$$\sum_{n,m} A^\dagger_m M_{mi}M_{jn} A_n=0\tag{7}$$
Which, again, I'm wondering if this can equal:
$$\sum_n \alpha_{ijn}A^T_nA_n=0\tag{8}$$
For a bunc of coefficients $\alpha_{ijn}$
I've been trying very hard to do it but mathematics has thwarted my every attempt, I think I'm up against some theorem that I'm neither aware of nor understand. Please solve this problem so I can put this to rest, cheers.
I'm not sure about a general $N\times N$ proof, but as far as proof-of-concept goes, there is the Lagrangian
$ L =\displaystyle\frac{m}{2}\begin{bmatrix}\dot{x}&\dot{y}\end{bmatrix}\begin{bmatrix}1&1\\-1&1\end{bmatrix}\begin{bmatrix}\dot{x}\\\dot{y}\end{bmatrix} -\frac{k}{2}\begin{bmatrix}x&y\end{bmatrix}\begin{bmatrix}1&1\\-1&1\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix} $
leaving aside questions of whether or not negative entries in the mass matrix are physically sensible. Substituting this into Lagrange's equations of motion leads to two decoupled simple harmonic oscillator equations
$ \begin{align} \ddot{x}+\omega^2x=0\\ \ddot{y}+\omega^2y=0 \end{align} $
with $\omega^2=k/m$.