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

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

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

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

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

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 $\widehat{A}$