Decoupling by diagonalization

We will solve the inhomogeneous ODE system $\dot{\mathbf x} = A\mathbf x + \mathbf q$ .

First diagonalize $A=V\Lambda V^{-1}$ . Then substitute $V \mathbf y = \mathbf x$ in the ODE system to find

\begin{align*} V \dot{\mathbf y} &= AV\mathbf y + \mathbf q \\ V \dot{\mathbf y} &= (V \Lambda V^{-1}) V \mathbf y + \mathbf q. \end{align*}

Multiply by $V^{-1}$

\begin{align*} \dot{\mathbf y} &= \Lambda \mathbf y + V^{-1} \mathbf q. \tag{1} \end{align*}

Since $\Lambda$ is a diagonal matrix, (1) is decoupled, so each variables derivative is a function of only that variable. Solve (1) for $\mathbf y$ , then compute $\mathbf x = V \mathbf y$ to reach the final solution.