1. Homogeneous Linear Systems
Thm. Let λ1, λ2, λ3 be distinct eigenvalues of A with corresponding eigenvectors K1, K2, K3. Then the general solution to X  = AX is

2. Ex. Solve

4. Ex. Solve

6. When eigenvalues are not distinct, things change
If λ1 repeats twice and has two eigenvectors, the solution is

7. Ex. Solve

9. If λ repeats twice and has eigenvector K, the solutions are and where P is given by

10. Ex. Solve

12. If λ repeats three times and has eigenvector K, the solutions are ,
, where Q is given by