(a) Suppose that $A$ is a real $n \times n$ matrix, and $\boldsymbol{w} \in \mathbb{R}^{n}$ and $\lambda_{1} \in \mathbb{R}$ are given so that $A \boldsymbol{w}=\lambda_{1} \boldsymbol{w}$. Further, let $S$ be a non-singular matrix such that $S \boldsymbol{w}=c e^{(1)}$, where $e^{(1)}$ is the first coordinate vector and $c \neq 0$.

Let $\widehat{A}=S A S^{-1}$. Prove that the eigenvalues of $A$ are $\lambda_{1}$ together with the eigenvalues of the bottom right $(n-1) \times(n-1)$ submatrix of $\widehat{A}$.

Explain briefly how, given a vector $\boldsymbol{w}$, an orthogonal matrix $S$ such that $S \boldsymbol{w}=c e^{(1)}$ can be constructed.

(b) Suppose that $A$ is a real $n \times n$ matrix, and two linearly independent vectors $\boldsymbol{v}, \boldsymbol{w} \in \mathbb{R}^{n}$ are given such that the linear subspace $L\{\boldsymbol{v}, \boldsymbol{w}\}$ spanned by $\boldsymbol{v}$ and $\boldsymbol{w}$ is invariant under the action of $A$, i.e.,

$$x \in L\{\boldsymbol{v}, \boldsymbol{w}\} \quad \Rightarrow \quad A x \in L\{\boldsymbol{v}, \boldsymbol{w}\}$$

Denote by $V$ an $n \times 2$ matrix whose two columns are the vectors $\boldsymbol{v}$ and $\boldsymbol{w}$, and let $S$ be a non-singular matrix such that $R=S V$ is upper triangular:

$$S V=S \times\left[\begin{array}{cc} v_{1} & w_{1} \\ v_{2} & w_{2} \\ v_{3} & w_{3} \\ : & : \\ v_{n} & w_{n} \end{array}\right]=\left[\begin{array}{cc} r_{11} & r_{12} \\ 0 & r_{22} \\ 0 & 0 \\ : & : \\ 0 & 0 \end{array}\right]$$

Again, let $\widehat{A}=S A S^{-1}$. Prove that the eigenvalues of $A$ are the eigenvalues of the top left $2 \times 2$ submatrix of $\widehat{A}$together with the eigenvalues of the bottom right $(n-2) \times(n-2)$ submatrix of $\widehat{A}$.

Explain briefly how, for given vectors $\boldsymbol{v}, \boldsymbol{w}$, an orthogonal matrix $S$ which satisfies (*) can be constructed.