the real distinct eigenvalues case
This is the easiest and most satisfying outcome of the eigenvalue method — the case where everything just works. The matrix A hands you n different real eigenvalues, each with its own eigenvector, and the general solution falls out as a simple sum with no extra effort.
Suppose A is n-by-n and its characteristic equation det(A - lambda I) = 0 has n real, distinct roots lambda_1, ..., lambda_n. Distinct eigenvalues automatically come with independent eigenvectors v_1, ..., v_n, so each pair gives a solution e^(lambda_i t) v_i, and these n solutions form a fundamental set straight away. The general solution is x(t) = c1 e^(lambda_1 t) v_1 + ... + cn e^(lambda_n t) v_n. The geometry is clean: along each eigenvector direction the motion is a pure straight-line solution that grows if its lambda is positive and decays if negative; a general solution is a blend of these straight-line motions, dominated in the long run by the term with the largest lambda.
Because every term is a real exponential times a fixed vector, there is no oscillation here — solutions either run off to infinity, collapse to the origin, or split (some directions growing, others decaying, giving a saddle). It is the textbook starting case and the one to master first; the complex and repeated cases are the two ways this clean picture can fail.
A = [3, 0; 0, -1] has eigenvalues 3 (eigenvector (1,0)) and -1 (eigenvector (0,1)). General solution x(t) = c1 e^(3t) (1, 0) + c2 e^(-t) (0, 1): the first coordinate blows up, the second decays — a saddle.
Distinct real eigenvalues: a sum of real exponentials, each along its own eigenvector — no rotation anywhere.
Distinct eigenvalues guarantee independent eigenvectors, so this case never needs generalized eigenvectors. The repeated-root case can still have a full set of eigenvectors, but it is not guaranteed — that is the subtlety distinctness avoids.