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

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

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

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  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