 Iteration matrix

For a stationary iteration $x^{(k+1)}=H{x}^{(k)}+v$, the matrix $H$ propagates the error by $e^{(k+1)}=H{e}^{(k)}$. 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 $A{x}_{\ast }=b$, subtracting its fixed point equation from the update givesThus $b$ 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.