##### Exercise10

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

1. \begin{equation*} \begin{aligned} A \amp {}={} \left[\begin{array}{rr} 1.2 \amp 0 \\ 0 \amp 0.7 \\ \end{array}\right] \\ \\ \xvec \amp {}={} \twovec{1}{2}\text{.} \end{aligned} \text{.} \end{equation*}
2. \begin{equation*} \begin{aligned} A \amp {}={} \left[\begin{array}{rr} 0.4 \amp 0 \\ 0 \amp 0.7 \\ \end{array}\right] \\ \\ \xvec \amp {}={} \twovec{3}{3}\text{.} \end{aligned} \end{equation*}
3. \begin{equation*} \begin{aligned} A \amp {}={} \left[\begin{array}{rr} 1.2 \amp 0 \\ 0 \amp 1.4 \\ \end{array}\right] \\ \\ \xvec\amp{}={}\twovec{1}{1}\text{.} \end{aligned} \end{equation*}
4. \begin{equation*} \begin{aligned} A \amp {}={} \left[\begin{array}{rr} 0.95 \amp 0.25 \\ 0.25 \amp 0.95 \\ \end{array}\right] \\ \\ \xvec\amp{}={}\twovec{1}{0}\text{.} \end{aligned} \end{equation*}

Find the eigenvalues and eigenvectors of $$A$$ to create your sketch.

5. If $$A$$ is a $$2\times2$$ matrix with eigenvalues $$\lambda_1=0.7$$ and $$\lambda_2=0.5$$ and $$\xvec$$ is any vector, what happens to $$A^k\xvec$$ when $$k$$ becomes very large?

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