Past exam of the mathematics course of the University of Cambridge / 2010 / ii / Paper 1 / 39A / b / Solution

Write $A=D+L+U$, with $D$ diagonal and $L,U$ the strict triangular parts. The Jacobi method updates all coordinates using the preceding iterate:For a symmetric positive definite matrix, $D$ has positive diagonal entries. In its splitting take $M=D$, $N=D-A$. The theorem requires $M^{\ast }+N=2D-A$ to be positive definite, which need not hold for every positive definite matrix.

For this tridiagonal matrix, let $S=\mathrm{diag}((-1{)}^1,\dots ,(-1{)}^n)$. Neighboring indices have opposite signs, so $SAS=2D-A$: the diagonal stays fixed and every nonzero off-diagonal entry changes sign. ThusThe Householder-John theorem now applies and proves Jacobi convergence for every initial vector. This is precisely why the tridiagonal structure matters.