Consider the linear system $A$x=b, where $A$ is a symmetric matrix. Suppose that $M-N$ is a splitting of $A$, where $M$ is symmetric positive definite and $N$ is symmetric. Show that if $\lambda_{min}(M)> \rho(N)$, then the iterative scheme $M$x$_{k+1}$=$N$x$_k+$b converges to xfor any initial guess x$_0$.
So I want to prove that if the minimum eigenvalue of $M$> maximum absolute eigenvalue of $N$, then $M$x$_{k+1}$=$N$x$_k+$b converges to x for all x$_0$.
I have no idea where to start. Some hints on where to start are appreciated. I prefer no full answers so I can try to finish the problem myself.
You have enough information to show $$ \|M^{-1}\|_2 < \|N\|_2^{-1}$$ This will imply that $$ \|M^{-1} N \|_2 < 1$$ which is enough to ensure convergence of the stationary iteration.
You will need to use the connection between eigenvalues and the 2-norm of symmetric matrices to derive the first inequality.