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} 1.4 \amp 0 \\ 0 \amp 0.7 \\ \end{array}\right] \\ \\ \xvec \amp {}={} \twovec{1}{2}\text{.} \end{aligned} \text{.} \end{equation*}

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  2. \begin{equation*} \begin{aligned} A \amp {}={} \left[\begin{array}{rr} 0.6 \amp 0 \\ 0 \amp 0.9 \\ \end{array}\right] \\ \\ \xvec \amp {}={} \twovec{4}{3}\text{.} \end{aligned} \end{equation*}

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  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{2}{1}\text{.} \end{aligned} \end{equation*}

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  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{3}{0}\text{.} \end{aligned} \end{equation*}

    Find the eigenvalues and eigenvectors of \(A\) to create your sketch.

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  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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