= Solution
One form of the <Householder-John theorem> is as follows. Suppose $A=A^*$ and $A=M-N$, with $M$ invertible and $W=M+M^*-A$ positive definite. Then the stationary iteration $Mx^{(k+1)}=Nx^{(k)}+b$ converges for every initial vector exactly when $A$ is positive definite; its iteration matrix is $B=M^{-1}N$ and convergence means <spectral radius> below one.
For a known positive definite system, design a splitting with $M$ easy to solve, such as diagonal or triangular, and arrange $M+M^*-A>0$. The theorem then guarantees convergence. For example $M=sI$ has $W=2sI-A>0$ when $2s$ exceeds the largest <eigenvalue> of $A$. In the converse direction, the discrete energy identity $A-B^*AB=(I-B)^*W(I-B)$ explains the criterion: stable $B$ gives $A$ as a convergent sum of positive matrices, while positive $A,W$ force every iteration <eigenvalue> inside the unit disk.
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