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

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

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

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

$$R=S V=S \times\left(\begin{array}{cc} v_{1} & w_{1} \\ v_{2} & w_{2} \\ \vdots & \vdots \\ v_{n} & w_{n} \end{array}\right)=\left(\begin{array}{cc} r_{11} & r_{12} \\ 0 & r_{22} \\ 0 & 0 \\ \vdots & \vdots \\ 0 & 0 \end{array}\right)$$

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 $\widehat{A} .$