A linear autonomous system has the form

$$\mathbf x'=A\mathbf x,$$

where $A$ is a constant matrix.

If $\mathbf v$ is an eigenvector of $A$ with eigenvalue $\lambda$, then

$$\mathbf x(t)=e^{\lambda t}\mathbf v$$

is a solution because

$$\mathbf x'=\lambda e^{\lambda t}\mathbf v=A\mathbf x.$$

When enough independent eigenvectors exist, the general solution is a linear combination of these eigenmodes.

The eigenvalues determine the time dependence: negative real parts produce decay, positive real parts produce growth, and nonzero imaginary parts produce oscillation.

This connects linear algebra directly to dynamics: eigenvectors identify independent modes of motion, while eigenvalues determine how each mode evolves.