We discuss the solution of Hermitian positive definite systems Ax=b by the preconditioned conjugate gradient method with a preconditioner M. In general, the smaller the condition number κ(MAM) is, the faster the convergence rate will be. For a given unitary matrix Q, let M = {Q*ΛQ | Λ is an n-by-n complex diagonal matrix} and M={Q*ΛQ | Λ is an n-by-n positive definite diagonal matrix}. The preconditioner M that minimizes κ(MAM) over M is called the best conditioned preconditioner for the matrix A over M. We prove that if QAQ* has Young's Property A, then M is nothing new but the minimizer of {norm of matrix}M -A{norm of matrix} over M. Here {norm of matrix} · {norm of matrix} denotes the Frobenius norm. Some applications are also given here.