Exercise9

For each of the following matrices, sketch the vector $$\xvec = \twovec{1}{0}$$ and powers $$A^k\xvec$$ for $$k=1,2,3,4\text{.}$$

1. $$A = \left[\begin{array}{rr} 0 \amp -1.4 \\ 1.4 \amp 0 \\ \end{array}\right] \text{.}$$

2. $$A = \left[\begin{array}{rr} 0 \amp -0.8 \\ 0.8 \amp 0 \\ \end{array}\right] \text{.}$$

3. $$A = \left[\begin{array}{rr} 0 \amp -1 \\ 1 \amp 0 \\ \end{array}\right] \text{.}$$

4. Consider a matrix of the form $$C=\left[\begin{array}{rr} a \amp -b \\ b \amp a \\ \end{array}\right]$$ with $$r=\sqrt{a^2+b^2}\text{.}$$ What happens when $$k$$ becomes very large when

1. $$r \lt 1\text{.}$$

2. $$r = 1\text{.}$$

3. $$r \gt 1\text{.}$$

in-context