Let $A \in \mathbb{R}^{n \times n}$ with $n>2$ and define $\operatorname{Spec}(A)=\{\lambda \in \mathbb{C} \mid A-\lambda I$ is not invertible $\}$.

The QR algorithm for computing $\operatorname{Spec}(A)$ is defined as follows. Set $A_{0}=A$. For $k=0,1, \ldots$ compute the $\mathrm{QR}$ factorization $A_{k}=Q_{k} R_{k}$ and set $A_{k+1}=R_{k} Q_{k}$. (Here $Q_{k}$ is an $n \times n$ orthogonal matrix and $R_{k}$ is an $n \times n$ upper triangular matrix.)

(a) Show that $A_{k+1}$ is related to the original matrix $A$ by the similarity transformation $A_{k+1}=\bar{Q}_{k}^{T} A \bar{Q}_{k}$, where $\bar{Q}_{k}=Q_{0} Q_{1} \cdots Q_{k}$ is orthogonal and $\bar{Q}_{k} \bar{R}_{k}$ is the QR factorization of $A^{k+1}$ with $\bar{R}_{k}=R_{k} R_{k-1} \cdots R_{0}$.

(b) Suppose that $A$ is symmetric and that its eigenvalues satisfy

$$\left|\lambda_{1}\right|<\left|\lambda_{2}\right|<\cdots<\left|\lambda_{n-1}\right|=\left|\lambda_{n}\right|$$

Suppose, in addition, that the first two canonical basis vectors are given by $\mathbf{e}_{1}=\sum_{i=1}^{n} b_{i} \mathbf{w}_{i}$, $\mathbf{e}_{2}=\sum_{i=1}^{n} c_{i} \mathbf{w}_{i}$, where $b_{i} \neq 0, c_{i} \neq 0$ for $i=1, \ldots, n$ and $\left\{\mathbf{w}_{i}\right\}_{i=1}^{n}$ are the normalised eigenvectors of $A$.

Let $B_{k} \in \mathbb{R}^{2 \times 2}$ be the $2 \times 2$ upper left corner of $A_{k}$. Show that $d_{\mathrm{H}}\left(\operatorname{Spec}\left(B_{k}\right), S\right) \rightarrow 0$ as $k \rightarrow \infty$, where $S=\left\{\lambda_{n}\right\} \cup\left\{\lambda_{n-1}\right\}$ and $d_{\mathrm{H}}$ denotes the Hausdorff metric

$$d_{\mathrm{H}}(X, Y)=\max \left\{\sup _{x \in X} \inf _{y \in Y}|x-y|, \sup _{y \in Y} \inf _{x \in X}|x-y|\right\}, \quad X, Y \subset \mathbb{C}$$

[Hint: You may use the fact that for real symmetric matrices $U, V$ we have $\left.d_{\mathrm{H}}(\operatorname{Spec}(U), \operatorname{Spec}(V)) \leqslant\|U-V\|_{2} \cdot\right]$