Second Order Matrix ODE

Given the equation $ \frac{d^2X}{dt^2} = MX$ and the appropriate initial values, how would one go about solving this equation? I’ve looked at Qualitative dependence of solution to second-order matrix differential equation on eigenvalues, which was very useful but I don’t really understand how the change of basis was performed nor how the eigenvalues could be found.

Are there any resources that explain how to tackle second order matrix equations out there?