Suppose we have the following system of coupled differential equations: $$x'=4x+4y\\ y'=x+y$$ One way to obtain a general solution of the system, for $x(t)$ and $y(t)$, is by seeking solutions in the form $\mathbf{x}(t)=\xi\exp(rt)$, where $\mathbf{x}(t)=[x(t),y(t)]^T$ and $\xi=[\xi_1,\xi_2]^T$.
An alternative way would be to use the diagonalization method. Setting $\mathbf{x}'=\mathbf{A}\mathbf{x}$ and making a change of variables $\mathbf{x}=\mathbf{Q}\tilde{\mathbf{x}}$. After some steps we'll get $\tilde{\mathbf{x}}'=\mathbf{D}\tilde{\mathbf{x}}$, where $\mathbf{D}$ is a diagonal matrix. After that we can solve the system for the new variables and at the end change back to the original variables.
Here are my questions. Which method is prefered for its rigor and efficiency? Which one is more appreciated from a mathematical perspective as it shows in a clearer way what's happening?
Thank you.
The two methods really amount to the same thing. In order for $\xi \exp(rt)$ to be a solution, you need $\xi$ to be an eigenvector of $A$ for eigenvalue $r$. The eigenvectors (assuming there's a complete set of them) form the columns of the matrix that you use to diagonalize $A$, and the eigenvalues are the diagonal entries of $D$.