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Subtract a steady state to obtain $e(t)=s(t)-s_{\ast }$ with $\dot{e}=Ae$, hence $e(t)=e^{At}e(0)$. The steady state attracts every initial condition exactly whenIn other words, $A$ must be a Hurwitz stable matrix. To see sufficiency without assuming diagonalizability, each block in its Jordan normal form contributes terms of the form $t^k{e}^{\mu t}$ with finite $k$. Negative real part makes every such term tend to zero, so the whole matrix exponential tends to zero.

For necessity, an eigenmode with positive real part grows, and a mode with zero real part persists or oscillates instead of decaying. For a complex eigenvalue of a real matrix, take the real and imaginary parts of its eigenvector to obtain real nondecaying solutions. Thus any eigenvalue in the closed right half-plane prevents convergence of all deviations. Physical density operators span the trace-one affine space, so convergence from all physical initial states also forces decay on every traceless direction.

This gives global asymptotic stability for the affine linear system and uniqueness of its steady state. A merely negative-semidefinite symmetric part proves only nonincrease of a particular norm, not this necessary-and-sufficient condition.

A steady state is attractive here if every physical initial density operator approaches it. Subtracting its Affine Bloch equation from the trajectory equation gives $\delta \dot{s}=A\delta s$, so $\delta s(t)=e^{At}\delta s(0)$. The necessary and sufficient condition isSuch an $A$ is a Hurwitz stable matrix. To prove sufficiency without assuming diagonalizability, put it in Jordan normal form. Each Jordan block contributes $e^{\lambda t}{\sum }_{j=0}^{q-1}{t}^j{N}^j/j!$, and the negative real part makes the exponential dominate all polynomial factors. For necessity, an eigenmode with positive real part grows; an eigenmode with zero real part remains constant or oscillates, so it does not tend to zero. A nontrivial Jordan block on the imaginary axis also prevents decay.

The physical Bloch vector body has nonempty interior, so differences of physical initial vectors from $s_{\ast }$ span the full real coordinate space. Thus convergence for every physical initial state also forces $e^{At}\to 0$ on that space. This rules out hiding a nondecaying eigenmode outside the physical set. The strict spectral condition entails uniqueness and global asymptotic stability of the steady state.