I have two coupled differential equations: $$\tag{1} \begin{array}{l} \dot{z}_{1}=-H_{11} z_{1}-H_{12} z_{2} \\ \dot{z}_{2}=-H_{21} z_{1} \end{array}$$ And I want to find the correlation functions: $$\tag{2} \left\langle z_{1}(t) z_{1}(0)\right\rangle, \quad\left\langle z_{2}(t) z_{2}(0)\right\rangle, \quad\left\langle z_{1}(t) z_{2}(0)\right\rangle$$
I rewrote the equations as a matrix equation to get: $$\begin{array}{l} z(t)=\left[\begin{array}{l} z_{1}(t) \\ z_{2}(t) \end{array}\right], \quad H=\left[\begin{array}{cc} H_{11} & H_{12} \\ H_{21} & 0 \end{array}\right] \\ \frac{d}{d t}\left\langle z_{i}(t) z_{j}(0)\right\rangle=-H_{i k}\left\langle z_{k}(t) z_{j}(0)\right\rangle \end{array}$$ Where $\left\langle z_{i}(t) z_{j}(0)\right\rangle$ is the correlation between $z_i(t)$ and $z_j(0)$. I am unfamiliar with matrix differential equations, and I am not sure how to solve for $\left\langle z_{i}(t) z_{j}(0)\right\rangle$?
The general solution to $\dot{X} = - H X$ is $X(t) = \exp(-tH) X(0)$, where $\exp(-tH)$ is a matrix exponential. If $H_{11}^2 + 4 H_{21} H_{12} \ne 0$, your $H$ has distinct eigenvalues $$ \lambda_{\pm} = \frac{H_{11}}{2} \pm \frac{\sqrt{H_{11}^2 + 4 H_{21} H_{12}}}{2}$$ and $$\exp(-tH) = e^{-t \lambda_+} \frac{H - \lambda_- I}{\lambda_+-\lambda_-} + e^{-t \lambda_-} \frac{H - \lambda_+ I}{\lambda_- - \lambda_+}$$