I'm learning about normal modes right now and I have a question pertaining to the way the equations of motion are written vis-a-vis summation notation. The book defines the equations of motion of a system of $n$ masses attached to springs, whose displacements are $q_j$ as the following: $$\underset{j}{\sum} (A_{jk}q_j + m_{jk}\ddot{q_{j}}) = 0$$My quandary is caused by the subscript $k$ that appears in the undetermined coefficients $A_{jk}$ as well as the masses $m_{jk}$. I just want to know what values of $k$ to use here in this equation in the general case. Since it's not explicitly stated, I'm sure there's an implicit understanding there.
Let's take for instance $n = 2$. So, we have $2$ masses attached to springs and $2$ solutions for the displacements (from equilibrium) $q_j$ of each mass. Then we must have $2$ equations of motion. I'm assuming $A$ and $m$ are $2 \times 2$ matrices and then is it the case - in this particular situation - that we have the relation $\delta_{jk}(A_{jk}q_j + m_{jk}\ddot{q_j})$? Such that our equations of motion become $$A_{11}q_1 + m_{11}\ddot{q_{1}} = 0\\ A_{22}q_2 + m_{22}\ddot{q_{2}} = 0$$
If this is true, then for the case $n \gt 2$ how do we treat the matrices $A$ and $m$? Do they just become $n \times n$ matrices? Where the elements on the diagonal are the nonzero ones?
Without knowing what book you're referencing, my interpretation is that the $k$ in the equation of motion represents the fact that there are $n$ equations being represented, indexed from $k = 1$ to $k = n$. So for 2 masses your equations are:
$$\begin{eqnarray} \sum_j A_{j1} q_j + m_{j1} \ddot{q}_j & = & 0 \\ \sum_j A_{j2} q_j + m_{j2} \ddot{q}_j & = & 0 \end{eqnarray}$$
You could also simplify these equations into a single matrix equation by writing $A \mathbf{q} + m \ddot{\mathbf{q}} = \mathbf{0}$.