Let $A$ be a singular and symmetric matrix, with $\lambda_1=0$ and $\lambda_i >0$ for $i=2,\ldots,n$.
Consider the iteration
$$x^{*} = b- (A-I)x_k$$
$$x_{k+1} = \alpha x^{*} +(1- \alpha)x_k$$Under which conditions on $x_0$, $\alpha$ and $b$, does it converge to the true solution of $Ax =b$?
I really can't move. I tried to compute $e_{k+1}$ but I couldn't find any useful relation. Also, I don't know how to find some constraints on $x_0$.
EDIT
I tried to follow @uranix comment's and I found: $$e_{k+1} = \alpha b + (I – \alpha A) x_k -x $$
which I rewrite (using consistency) as $$e_{k+1} = (I-\alpha A)(x_k -x)=(I- \alpha A)e_{k}$$
Therefore $$e_{k+1} = (I-\alpha A)^k e_0$$
Now I would require the spectral radius to be less than $1$, but since $$\lambda(I -\alpha A)= 1-\lambda(A)$$ I have that the first eigenvalue is $1-\alpha \lambda_1=1-\alpha \cdot 0 = 1$
So I can't say anything about convergence… there must be another way. Indeed, I didn't use symmetry and also no conditions on $x_0$, as writen in the text
Best Answer
A little hint.
As I said in comments, consider the eigenvalue basis. The basis vectors are orthogonal and may be scaled to form an orthonormal basis: $$ A \phi_m = \lambda_m \phi_m, \quad m = 1, \dots, m\\ (\phi_m, \phi_{m'}) = \delta_{mm'}. $$
Expanding error vectors over the basis $e_k = \sum_{m=1}^n c_{k,m} \phi_m$ allows to rewrite convergence condition using expansion coefficients. Using Parseval's identity $$ \|e_k\|_2^2 = \sum_{m} c_{k,m}^2 $$ we obtain that $e_k \to 0$ happens only if for all $m$ each coefficient converges to zero, that is $$ \lim_{k \to \infty} c_{k,m} = 0, \quad m = 1,\dots,n. $$
Acting with $(I - \alpha A)^k$ on $e_0$ acts on each eigenvalue separately: $$ e_k = (I - \alpha A)^k e_0 = (I - \alpha A)^k \sum_{m=1}^n c_{0,m} \phi_m = \\ = \sum_{m=1}^n c_{0,m} (I - \alpha A)^k \phi_m = \sum_{m=1}^n c_{0,m} (1 - \alpha \lambda_m)^k \phi_m. $$
Comparing the right hand side with $\sum_{m=1}^n c_{k,m} \phi_m$ we immediately get the relation $$ c_{k,m} = (1 - \alpha \lambda_m)^k c_{0,m}. $$
Now it's up to you to find the conditions when $\lim_{k \to \infty} c_{k,m} = 0$ for every $m = 1,\dots,n$.