I have the following system of homogeneous linear differential equations that I wish to solve: $$ \textbf{M}\ddot{\textbf{q}} + \textbf{G}\dot{\textbf{q}} + \textbf{D}\dot{\textbf{q}} + \textbf{K}\textbf{q} = \textbf{0} $$
In this case the vector $\textbf{q}$ is a $\boldsymbol{4 \times 1}$ vector of the system states. Let's say $\textbf{q} = \begin{bmatrix} q_{1} & q_{2} & q_{3} & q_{4} \end{bmatrix}^\top$.
The matrices involved are as follows: $$ \textbf{M} = \begin{bmatrix} m_{1} & 0 & 0 & 0 \\ 0 & m_{2} & 0 & 0 \\ 0 & 0 & m_{3} & 0 \\ 0 & 0 & 0 & m_{4} \\ \end{bmatrix} $$
$$ \textbf{D} = \begin{bmatrix} d_{11} & d_{12} & 0 & 0 \\ d_{21} & d_{22} & 0 & 0 \\ 0 & 0 & d_{33} & d_{34} \\ 0 & 0 & d_{43} & d_{44} \\ \end{bmatrix} $$
$$ \textbf{K} = \begin{bmatrix} k_{11} & k_{12} & 0 & 0 \\ k_{21} & k_{22} & 0 & 0 \\ 0 & 0 & k_{33} & k_{34} \\ 0 & 0 & k_{43} & k_{44} \\ \end{bmatrix} $$
$$ \textbf{G} = \begin{bmatrix} 0 & 0 & g_{1} & 0 \\ 0 & 0 & 0 & 0 \\ g_{2} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{bmatrix} $$
By introducing the state vector $ \textbf{x(t)} = \pmatrix{\textbf{q(t)} \\ \dot{\textbf{q(t)}}}$, the system is transformed into the state space form $ \dot{\textbf{x(t)}} = \textbf{A}\textbf{x(t)} + \textbf{B}\textbf{u(t)}$.
In this case, I end up with the $\textbf{A}$ matrix: $$ \textbf{A} = \begin{bmatrix} \textbf{0} & \textbf{I} \\ -\textbf{M}^{-1}[\textbf{K} + \textbf{D}] & -\textbf{M}^{-1}\textbf{G} \end{bmatrix} $$
I can then solve for the eigenvectors and eigenvalues of $\textbf{A}$. I just wanted to ask, what do these eigenvectors and eigenvalues represent? And how do they relate to the individual state variables $\textbf{q} = \begin{bmatrix} q_{1} & q_{2} & q_{3} & q_{4} \end{bmatrix}^\top$?
I am a bit confused, especially considering that this is a coupled system of ODEs. I want to understand how the different parameters involved affect the dynamics of the system.