Due: November 27th, 2017
Math 18 Assignment 7
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For each of the following matrices, given the eigenvalues, find a basis for each of the corresponding eigenspaces.
 $\begin{bmatrix}2&2\\2&1\end{bmatrix}$ has eigenvalues $3$ and $2$.
 $\begin{bmatrix}4&1&0&0\\0&4&0&0\\0&0&4&0\\0&0&16&4\end{bmatrix}$ has only the eigenvalue $4$.
 $\begin{bmatrix}19&8&18&14\\24&9&37&31\\24&12&6&9\\24&12&8&11\end{bmatrix}$ has eigenvalues $3, 2,$ and $1$.

Consider the matrix
$$Q = \begin{bmatrix}2&2&2&2\\2&2&2&2\\2&2&2&2\\2&2&2&2\end{bmatrix}.$$
 Find a nonzero eigenvalue and a corresponding eigenvector for $Q$.
 Find the rank of $Q$.
 Without further computation, find the characteristic polynomial of $Q$.

Suppose that $A^2$ is the zero matrix.
Show that $0$ is an eigenvalue of $A$, and explain why $A$ can have no other eigenvalues.

Suppose that $0 \leq \theta < 2\pi$, let $R$ be the matrix
$$R = \begin{bmatrix}\cos\theta&\sin\theta\\\sin\theta&\cos\theta\end{bmatrix}.$$
 For what values of $\theta$ does $R$ have a nontrivial (real) eigenvector?
 Notice that $R$ corresponds to rotation by an angle $\theta$ about the origin; explain why your answer above makes sense geometrically.
 Consider the following matrices:
$$A = \begin{bmatrix}2&1\\0&2\end{bmatrix} \hspace{2in} B = \begin{bmatrix}2&0\\0&2\end{bmatrix}.$$
 Find the characteristic polynomials of $A$ and $B$. (This requires very little work.)
 Find a nontrivial eigenvector for $A$, and two linearly independent eigenvectors for $B$.
 Is there an eigenvector of $A$ which is linearly independent from the one you found in Part B? If so, find one; if not, explain why not.
 Let $M$ be the following matrix:
$$M = \begin{bmatrix}6&4&2\\5&3&1\\2&2&2\end{bmatrix}.$$
 Find the eigenvalues of $M$.
 Find a linearly independent set of three eigenvectors of $M$.
 Write down matrices $P$ and $D$ so that $D$ is diagonal, $P$ is invertible, and $M = PDP^{1}$.
 Compute $M^3$.