Iterative methods to solve linear system of equations

[Sam80]

Show that if (Spectral radius) ρ(M)≥1 then there are x₀ and c such that the iteration x_k+1 = Mx_k+c fails to converge

I am very lost on how to solve this problem. All the literature I have been able to read up proves the converse of the statement. I am unable to crack this.

Show that if A is symmetric & positive definite then B_SGS (Symmetric Gauss Seidel) is also symmetric and positive definite

Any help will be greatly appreciated.

Sam

 

[LightHero]

uel's Answer:Let M be a matrix with spectral radius ρ(M) ≥ 1. We then have the iteration xk+1 = Mxk + c for some c in R.If this iteration fails to converge, then it must diverge to infinity.Assume, to reach a contradiction, that this iteration does converge.Then there exists constants K and N such that for all n > N, |xn+1 - xn| <= K.However, since ρ(M) >= 1, we have that |Mxn + c| >= |Mxn| - |c| > |Mxn| - K for all n > N.But this contradicts the assumption that |xn+1 - xn| <= K for all n > N.Therefore, if ρ(M) ≥ 1, then there exists x0 and c such that the iteration xk+1 = Mxk + c fails to converge.