decoupling by diagonalization
A coupled system is hard precisely because each unknown's fate is tangled up with the others. But what if you could find a clever new set of coordinates in which the equations simply fall apart — each new variable evolving entirely on its own, ignoring the rest? When the matrix is diagonalizable, you can, and the technique is called decoupling by diagonalization.
Suppose A has n independent eigenvectors. Build a matrix P whose columns are those eigenvectors; then P^(-1) A P = D, a diagonal matrix carrying the eigenvalues. Change variables with y = P^(-1) x (so x = P y). Substituting into x' = A x and multiplying by P^(-1) gives y' = D y. Because D is diagonal, this is n separate scalar equations: y1' = lambda_1 y1, y2' = lambda_2 y2, and so on — utterly uncoupled, each solved instantly as y_i = c_i e^(lambda_i t). Finally transform back with x = P y to recover the original unknowns. The new coordinates y are simply the components of x along the eigenvector directions, which is exactly why the coupling vanishes in them.
Diagonalization is the conceptual heart of the eigenvalue method — it explains why eigenvectors matter at all. In eigenvector coordinates the system is genuinely a stack of independent one-dimensional problems, the simplest objects in the whole theory. Its honest limit: when A is defective (a repeated eigenvalue short of eigenvectors) it cannot be fully diagonalized, and the best you can do is the nearly-diagonal Jordan form, where small coupling survives in the t-terms.
For A = [1, 2; 0, 3] with eigenvectors (1,0), (1,1), set P = [1, 1; 0, 1]; then P^(-1) A P = [1, 0; 0, 3] = D. The system becomes y1' = y1, y2' = 3 y2, solved as y1 = c1 e^t, y2 = c2 e^(3t); then x = P y.
In eigenvector coordinates the matrix is diagonal, so the equations no longer talk to each other.
Decoupling by diagonalization works only when A has a full set of n independent eigenvectors. A defective matrix is not diagonalizable — the most you get is the Jordan form, where a residual coupling stays inside each Jordan block.