Relations between condition numbers and the convergence of the Jacobi method for real positive definite matrices

Summary

LetA be ann×n real symmetric diagonal dominant matrix with positive diagonal partD, and letS ²=D ⁻¹ andH=SAS. The following relation between the condition number κ(H)=‖H ⁻¹‖ ‖H‖ and the spectral radiusr of the Jacobi matrix associated toA is proved:

$$(\kappa (H) - 1)/(\kappa (H) + 1) \leqq r \leqq (\kappa (H) - 1)/(1 + \kappa (H)/(n - 1)).$$

Moreover, relations among κ(H), κ(A), the condition numbersC(H)=‖|H ⁻¹| |H|‖, andC(A) are investigated.