= Iteration matrix
{title2=$H$}
= Stationary iteration matrix
{synonym}
For a stationary iteration $x^{(k+1)}=Hx^{(k)}+v$, the matrix $H$ propagates the error by $e^{(k+1)}=He^{(k)}$. Convergence from every initial error is equivalent to $\rho(H)<1$.
For a <matrix splitting> $A=M-N$, the <iteration matrix> is $H=M^{-1}N=I-M^{-1}A$, where $I$ is the <identity matrix>. If $Ax_*=b$, subtracting its <fixed point> equation from the update gives
$$
e^{(k)}=x^{(k)}-x_*,\qquad e^{(k+1)}=He^{(k)},\qquad e^{(k)}=H^ke^{(0)}.
$$
Thus $b$ determines the <fixed point>, while $H$ determines how the <forward error> moves. An <eigenvector> $v$ with <eigenvalue> $\lambda$ is an error direction whose coefficient is multiplied by $\lambda$ on every step. A negative real <eigenvalue> reverses its sign; a zero <eigenvalue> removes that component in one step.
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