Linear Algebra with Differential Equations/Homogeneous Linear Differential Equations/Real, Distinct Eigenvalues Method

If the eigenvalues for the characteristic equation are real and distinct, mathematically, nothing is really wrong. Thus, by our guess and the existence and uniqueness theorem, for an n-size square matrix, the solution set is determined by:

$$\mathbf{X} = \{ e^{\lambda_1 \cdot t} ; e^{\lambda_2 \cdot t} ; ... ; e^{\lambda_{n-1} \cdot t} ; e^{\lambda_n \cdot t} \}$$

Then since the linear combination of two solutions is also a solution (which can be verified directly from the structure of the problem), we can form the general solution as such:

$$\mathbf{X} = c_1 e^{\lambda_1 \cdot t} + c_2 e^{\lambda_2 \cdot t} + ... + c_{n-1} e^{\lambda_{n-1} \cdot t} + c_n e^{\lambda_n \cdot t}$$

What's interesting is when the eigenvalues are not so simple.