= Solution
<Strict diagonal dominance> gives
$$
|b_{ii}|>\sum_{j\ne i}|b_{ij}|\geq0.
$$
Thus every $b_{ii}\ne0$, and the diagonal <matrix> $D$ has inverse $D^{-1}=\operatorname{diag}(1/b_{11},\ldots,1/b_{nn})$. The iteration <matrix> has entries
$$
t_{ii}=0,\qquad t_{ij}=\frac{b_{ij}}{b_{ii}}\quad(i\ne j).
$$
Each row <Gershgorin disc> for $T$ is centred at zero and has radius
$$
\rho_i=\sum_{j\ne i}|t_{ij}|=\frac{\sum_{j\ne i}|b_{ij}|}{|b_{ii}|}<1.
$$
Therefore every <eigenvalue> of $T$ satisfies
$$
\boxed{|\lambda|\leq\max_i\rho_i<1.}
$$
Equivalently, the induced <matrix norm> satisfies $\|T\|_\infty<1$, bounding the <spectral radius>. This is the usual <Jacobi convergence from strict diagonal dominance>; the Jacobi error <matrix> is $-T$, with the same eigenvalue magnitudes.
Back to article page
