Relations between condition numbers and the convergence of the Jacobi method for real positive definite matrices (Q791274)

Let A be an \(n\times n\) real symmetric diagonal dominant matrix with positive diagonal part D, and let \(S^ 2=D^{-1}\) and \(H=SAS\). The following relation between the condition number \(k(H)=\| H^{- 1}\|\| H\|\) and the spectral radius r of the Jacobi matrix associated to A is proved: \((k(H)-1)/(k(H)+1)\leq r\leq(k(H)-1)/(1+k(H)/(n- 1)).\) Moreover, relations among k(H), k(A), the condition numbers \(C(A)=\|| A^{-1}| | A| \|\), and C(H) are investigated.