##### 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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