iteration method for inverse matrix a)
Consider the iterative scheme
\begin{equation}
x_{n+1} = x_n + c (Ax_n - I) \tag{1}
\end{equation}
When the process converges, show that this scheme (where $c$ is an appropriately chosen real
number) can be used to calculate $A^{-1}$ . 
b) 
for $A= \left(\begin{matrix}2&1\\1&2\end{matrix}\right)$
find the values of $c$ for which the process in equation $(1)$ converges.
Which value of $c$ leads to the fastest convergence?
Hopefully, someone would be able to solve the above question for me!!
Thx!!
Hopefully, some one would be able to solve the above questions for me?
Thanks very much!!!!!!!!!!!!
 A: Assume that your current approximation, $x_n$, is different from $A^{-1}$ by an error matrix, $\epsilon_n$. Now, substituting into the iteration equation, we have
$$
A^{-1}+\epsilon_{n+1} = A^{-1}+\epsilon_n+c\left(A(A^{-1}+\epsilon_n)-I\right)
$$
or
$$
\epsilon_{n+1} = \epsilon_n + c(I+A\epsilon
_n-I) = (I+cA)\epsilon_n
$$
To see that the iteration method approximates $A^{-1}$ when it converges, note that if $\epsilon_n=0$, then $\epsilon_{n+1}=0$.
To see what we need to do to ensure convergence, note that
$$
\epsilon_{n+m} = (I+cA)^m\epsilon_n
$$
As such, we seek a value of $c$ for which $I+cA$ has eigenvalues of magnitude less than $1$. For the given matrix $A$, the eigenvalues of $I+cA$ are $c+1$ and $3c+1$. As such, it can easily be confirmed that, for real $c$, we require that $-2/3<c<0$. To find the value with fastest convergence, you need to find the one for which the eigenvalue with the larger magnitude will be minimised in magnitude.
That, I'll leave to you, with just one hint - as you make the magnitude of one larger, the other will get smaller, within a certain range.
