Exercise7

Consider the matrix \(A=\left[\begin{array}{rr} 0.5 \amp 0.6 \\ -0.3 \amp 1.4 \\ \end{array}\right] \text{.}\)

  1. Find the eigenvalues of \(A\) and a basis for their associated eigenspaces.

  2. Suppose that \(\xvec_0=\twovec{11}{6}\text{.}\) Express \(\xvec_0\) as a linear combination of eigenvectors of \(A\text{.}\)

  3. Define the vectors

    \begin{equation*} \begin{aligned} \xvec_1 \amp {}={} A\xvec_0 \\ \xvec_2 \amp {}={} A\xvec_1 = A^2\xvec_0 \\ \xvec_3 \amp {}={} A\xvec_2 = A^3\xvec_0 \\ \vdots \amp {}={} \vdots \end{aligned} \text{.} \end{equation*}

    Write \(\xvec_1\text{,}\) \(\xvec_2\text{,}\) and \(\xvec_3\) as a linear combination of eigenvectors of \(A\text{.}\)

  4. What happens to \(\xvec_k\) as \(k\) grows larger and larger?

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