Eigenvalues and operator norm

Question

$A: \mathbb R^2 \to \mathbb R^2$ is $2 \times 2$ matrix with eigenvalues $\frac{2}{3}$ and $\frac{9}{5}$. Prove that there exists

1. a non-zero vector $v$ with $\|Av\|> 2\|v\|$, and
2. a non-zero vector $v$ with $\|Av\|<\frac{1}{2} \|v\|$.

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By defining a continuous function from the unit circle $S$ in the plane, (which is a compact set) to the real line, I conclude that the image contains the closed interval $[\frac{2}{3}, \frac{9}{5}]$. But how do I conclude that the image doesn’t contain $\frac{1}{2}$ and $2$? Help solicited.

Answer 1

Choose $A=\begin{bmatrix} {2 \over 3} & 1000 \\ 0 & {9 \over 5}\end{bmatrix}$.

Note that $\|A (0,1)^T\| > 1000$.

Let $v=(\sqrt{1-{1 \over 1000^2}}, -{1 \over 1000^2})^T$, then and note that $Av = ({2\over 3} \sqrt{1-{1 \over 1000^2}}-1, {9 \over 5000})^T$ and $\|Av\| \le \sqrt{{1\over 3^2} + {1 \over 10^2}} < {11 \over 30} < { 1\over 2}$.