Jacobi convergence from strict diagonal dominance ($\Vert B{\Vert }_{\infty }={\max }_i{\sum }_{j\ne i}|a_{ij}/a_{ii}|<1$)

A strictly diagonally dominant matrix guarantees convergence of the Jacobi method for every initial vector.^[ref]

Indeed, using the maximum norm $\Vert v{\Vert }_{\infty }={\max }_i|v_i|$, the induced operator norm is the largest absolute row sum. The Jacobi method has $b_{ii}=0$ and $b_{ij}=-a_{ij}/a_{ii}$ for $j\ne i$, soThis gives a cheap sufficient check without computing any eigenvalues. For the Jacobi iteration for a three-variable tridiagonal system, the absolute row sums of $B$ are $1/4,1/2,1/4$, giving $\Vert B{\Vert }_{\infty }=1/2$. The true spectral radius $\sqrt{2}/4$ gives a sharper eventual rate.

Strict diagonal dominance is sufficient, not necessary: $A=\left(\begin{array}{cc}1& 2\\ 1/10& 1\end{array}\right)$ violates it in the first row, but its Jacobi method has eigenvalues $\pm 1/\sqrt{5}$ and converges. Conversely, being a positive-definite matrix alone is insufficient: the Jacobi convergence for a three-by-three equicorrelation matrix with $\alpha =3/4$ has positive eigenvalues $5/2,1/4,1/4$ for $A$, but spectral radius $3/2$ for its iteration matrix. The extra tridiagonal matrix structure in Jacobi convergence for a symmetric positive-definite tridiagonal matrix is therefore substantive.