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?
Best Answer
The idea is to simplify the equation by transforming it to the form $$ \frac{dY} {dt} = D Y $$ where $D$ is a constant diagonal matrix. This matrix equation is can be easily rewritten as a system of equations where each equation only features a single element of $Y$: $$ \frac{dY_{ij}} {dt} = D_{ii} Y_{ij} \quad \forall i, j $$ The problem is, what systems of the form $$ \frac{dX} {dt} = M X $$ where $M$ is a constant matrix, can transform variables so that we get an equation of the simplified firm above? Well, if $M$ is a matrix with a complete set of right eigenvectors which form a matrix $R$, then the corresponding matrix of left eigenvectors is $R^{-1}$ and the diagonal matrix of eigenvalues is $\Lambda=R^{-1}MR$. So, how do we get a nice diagonal matrix in our equation? If we premultiply by $R^{-1} $ and insert a factor of $I=RR^{-1}$ between $M$ and $X$ then $$ R^{-1} \frac{dX} {dt} = R^{-1} M R R^{-1} X $$ $$ \frac{dR^{-1}X} {dt} = \Lambda R^{-1} X $$ Using a letter to the repeated combination, $\eta=R^{-1} X$, yields $$ \frac{d\eta} {dt} = \Lambda \eta $$ which is in the simplified form desired.