Let $P \in \mathbb R^{n \times n}$ be a symmetric positive definite matrix with eigenvalues denoted by $\lambda_i$ and corresponding eigenvectors denoted by $v_i$. For each $j \in \{1, 2, 3, 4\}$, let $\alpha_j$ be a non-zero real number. Let $x: [0, \infty) \rightarrow \mathbb R^{2n}$ be a continuous, differentiable function satisfying \begin{align*} &\frac{d}{dt}x(t) = Ax(t), \\ &A = \left[\begin{array}{cc} \alpha_1P & \alpha_2I \\ \alpha_3P & \alpha_4I \end{array}\right] \in \mathbb R^{2n \times 2n}, \\ &x(0) = z \in \mathbb R^{2n}, \end{align*} where $I \in \mathbb R^{n \times n}$ is the identity matrix. What methods can be used to manually obtain the solution $x(t)$ to the above differential equation, expressed in terms of $\lambda_i$ and $v_i$? We may manipulate $z$ for convenience (for example, as done below).

If we don't require the $\alpha_j$ to be non-zero, the case $$A = \left[\begin{array}{cc} 0 & I \\ -P & 0 \end{array}\right]$$ gives rise to the following solution. Define $x_1 \in \mathbb R^n$ and $x_2 \in \mathbb R^n$ such that $$x = \left[\begin{array}{c} x_1 \\ x_2 \end{array}\right],$$ and let $$x_1(0) = \sum_{i=1}^n c_iv_i,$$ where each $c_i$ is a real number. I've been informed that, in this case, $$x(t) = \left[\begin{array}{c} \sum_{i=1}^n c_iv_i\cos\left(\sqrt{\lambda_i} t\right) \\ -\sum_{i=1}^n c_iv_i\sqrt{\lambda_i}\sin\left(\sqrt{\lambda_i} t\right) \end{array}\right].$$