Suppose we have a matrix differential equation in $\vec{x}(t)=\left(\begin{smallmatrix}x_{1}(t) \\ \vdots \\ x_{n}(t)\end{smallmatrix}\right)$, such that:
$$\frac{\mathrm{d}^{2}\vec{x}}{\mathrm{d}t^{2}}=-\mathbf{M}\vec{x}$$
And $\mathbf{M}$ is a real, symmetric matrix, which therefore diagonalizes to $\mathbf{M}=\mathbf{P}\mathbf{\Lambda}\mathbf{P}^{-1}$, with $\mathbf{\Lambda}=\operatorname{diag}(\lambda_{1},\dots,\lambda_{n})$. We can perform a change of basis such that $\vec{x}(t)=\mathbf{P}\vec{\eta}(t)$, and therefore we have the following:
$$\frac{\mathrm{d}^{2}\vec{\eta}}{\mathrm{d}t^{2}}=-\mathbf{\Lambda}\vec{\eta} \implies \frac{\mathrm{d}^{2}\eta_{i}}{\mathrm{d}t^{2}}=-\lambda_{i}\eta_{i}$$
This is easy to solve and gives us:
$$\eta_{i}=\begin{cases}A_{i}\cos(\sqrt{\lambda_{i}}t)+B_{i}\sin(\sqrt{\lambda_{i}}t) & \lambda_{i} > 0 \\ A_{i}+B_{i}t & \lambda_{i} = 0 \\ A_{i}e^{\sqrt{-\lambda_{i}}t}+B_{i}e^{-\sqrt{-\lambda_{i}}t} & \lambda_{i} < 0\end{cases}$$
We can therefore solve for $\vec{x}$ by computing $\mathbf{P}\vec{\eta}$. However, I am then asked how the qualitative behaviour of $\vec{x}$ depends upon the eigenvalues of $\mathbf{M}$. I am having trouble understanding what is required of me; we have the qualitative behaviour of $\eta_{i}$ and it's dependence upon $\lambda_{i}$, but without more explicit knowledge of the change of basis matrix $\mathbf{P}$ I'm not sure what we can say about $\vec{x}$?
The question then requires us to focus on the $2$-dimensional coupled system of differential equations:
\begin{align} \frac{\mathrm{d}^{2}x_{1}}{\mathrm{d}t^{2}}&=-kx_{1}-lx_{2} \\ \frac{\mathrm{d}^{2}x_{2}}{\mathrm{d}t^{2}}&= -lx_{1} - kx_{2} \end{align}
We can therefore construct $\mathbf{M}=\left(\begin{smallmatrix}k & l \\ l & k\end{smallmatrix}\right)$, which diagonalizes to give $\mathbf{\Lambda}=\operatorname{diag}(k-l,k+l)$ and $\mathbf{P}=\frac{1}{\sqrt{2}}\left(\begin{smallmatrix}-1 & 1 \\ 1 & 1\end{smallmatrix}\right)$. Now clearly the solution can be worked out explicitly using the method above and then the qualitative behaviour can be explored, but is there anything we can say without explicit calculation?