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

Let A be an n x n real symmetric diagonal dominant matrix with positive diagonal part D, and let S^2 = D, 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) ? r ? (k(H) -1)/(1 + k(H)/(n -1)). Moreover, relations among k(H), k(A), the condition numbers C(H)=?| H^(-1)||H|?, and C(A) are investigated.