## Prove that the Length $\|A^n\mathbf{v}\|$ is As Small As We Like.

## Problem 381

Consider the matrix

\[A=\begin{bmatrix}

3/2 & 2\\

-1& -3/2

\end{bmatrix} \in M_{2\times 2}(\R).\]

**(a)** Find the eigenvalues and corresponding eigenvectors of $A$.

**(b)** Show that for $\mathbf{v}=\begin{bmatrix}

1 \\

0

\end{bmatrix}\in \R^2$, we can choose $n$ large enough so that the length $\|A^n\mathbf{v}\|$ is as small as we like.

(*University of California, Berkeley, Linear Algebra Final Exam Problem*)

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