Exercise 7

Consider the matrix $A=\left[\begin{array}{rr} 0.5 & 0.6 \\ -0.3 & 1.4 \\ \end{array}\right]$.

1. Find the eigenvalues of $A$ and a basis for their associated eigenspaces.

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

3. Define the vectors $$\begin{aligned} \xvec_1 & {}={} A\xvec_0 \\ \xvec_2 & {}={} A\xvec_1 = A^2\xvec_0 \\ \xvec_3 & {}={} A\xvec_2 = A^3\xvec_0 \\ \vdots & {}={} \vdots \end{aligned}$$. Write $\xvec_1$, $\xvec_2$, and $\xvec_3$ as a linear combination of eigenvectors of $A$.

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