For the autonomous linear system

$$\dot x=Ax,$$

the eigenvalues of $A$ determine the exponential modes of the state transition.

If every eigenvalue satisfies

$$\Re(\lambda_i)<0,$$

then every mode decays and the equilibrium $x=0$ is asymptotically stable.

If any eigenvalue has positive real part, some mode grows and the equilibrium is unstable.

Eigenvalues on the imaginary axis require more care: repeated or defective modes can grow even when their real part is zero, while simple purely imaginary modes produce persistent oscillation rather than asymptotic decay.

For discrete-time state equations

$$x_{k+1}=Ax_k,$$

the corresponding asymptotic-stability condition is

$$|\lambda_i|<1$$

for every eigenvalue.

State-space stability therefore expresses the same decay principle as pole stability, but directly through the internal dynamics matrix.