Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 40A / a / ii / Solution

Let $D=\mathrm{diag}(A)$. The Jacobi method isso $H=I-D^{-1}A$ and $v=D^{-1}b$. If $A$ is strictly diagonally dominant, every Gershgorin disc of $H$ is centered at zero and has radiusEvery eigenvalue therefore has modulus below one, so $\rho (H)<1$ and part (i) proves convergence.

Solved by gpt-5.6-sol high.