For the homogeneous linear state equation

$$\dot x(t)=Ax(t),$$

the state evolves as

$$x(t)=e^{At}x(0),$$

where the matrix exponential is defined by the convergent series

$$e^{At}=I+At+\frac{A^2t^2}{2!}+\frac{A^3t^3}{3!}+\cdots.$$

The matrix

$$\Phi(t)=e^{At}$$

is the state-transition matrix: it maps an initial state to the state reached after time $t$ with zero input.

With an input,

$$\dot x=Ax+Bu,$$

the solution becomes

$$x(t)=e^{At}x(0)+\int_0^t e^{A(t-\tau)}Bu(\tau),d\tau.$$

The first term is the natural response and the integral is the forced response.

If $A$ is diagonalizable, its eigenvectors decouple the evolution into scalar exponential modes $e^{\lambda_i t}$. This connects state-space dynamics directly to the eigenstructure of $A$.