I have some trouble solving the following (2.36 from Vector Calculus by Hubbard):
Let $A$ be an $n \times n$ diagonal matrix: $A = \begin{bmatrix}\lambda_1 & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & \lambda_n\end{bmatrix}$, and suppose that one of the diagonal entries $\lambda_k$ satisfies $\inf_{k \ne j} |\lambda_k - \lambda_j| \ge m > 0$ for some $m$. Let $B$ be an $n \times n$ matrix. Find a number $R$, depending on $m$, such that if $|B| < R$, then Netwon's method will converge if it is used to solve $$ (A+B)\mathbf{x} = \mu\mathbf{x}, \quad \text{for } \mathbf{x} \text{ satisfying } |\mathbf{x}|^2 = 1, $$ starting at $\mathbf{x}_0 = \vec{e}_k$, $\mu_0 = \lambda_k$.
I have computed the derivative of the function $F\begin{pmatrix}\mathbf{x} \\ \mu\end{pmatrix} = \begin{pmatrix}(A+B-\mu I)\mathbf{x} \\ |\mathbf{x}|^2 - 1 \end{pmatrix}$ $$ \left[\mathbf{DF}\begin{pmatrix}\mathbf{x} \\ \mu\end{pmatrix}\right] = \begin{bmatrix}\lambda_1 + b_{11} - \mu & b_{12} & \cdots & b_{1n} & -x_1 \\ b_{21} & \lambda_2 + b_{22} - \mu & \cdots & b_{2n} & -x_2 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ b_{n1} & b_{n2} & \cdots & \lambda_n + b_{nn} - \mu & -x_n \\ 2x_1 & 2x_2 & \cdots & 2x_n & 0 \end{bmatrix}$$,
and its Lipschitz ratio $M = \sqrt{5n}$. However, I don't know how to compute the length of the inverse of the derivative. I tried row reducing it but it quickly becomes out of hand.
