Due: November 27th, 2017

Math 18 Assignment 7

Answers to these problems are now available here.

  1. For each of the following matrices, given the eigenvalues, find a basis for each of the corresponding eigenspaces.
    1. $\begin{bmatrix}2&2\\2&-1\end{bmatrix}$ has eigenvalues $3$ and $-2$.
    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 $4$.
    3. $\begin{bmatrix}19&8&-18&14\\-24&-9&37&-31\\24&12&6&-9\\24&12&8&-11\end{bmatrix}$ has eigenvalues $3, -2,$ and $1$.
  2. Consider the matrix $$Q = \begin{bmatrix}2&2&2&2\\2&2&2&2\\2&2&2&2\\2&2&2&2\end{bmatrix}.$$
    1. Find a non-zero eigenvalue and a corresponding eigenvector for $Q$.
    2. Find the rank of $Q$.
    3. Without further computation, find the characteristic polynomial of $Q$.
  3. 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.
  4. Suppose that $0 \leq \theta < 2\pi$, let $R$ be the matrix $$R = \begin{bmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{bmatrix}.$$
    1. For what values of $\theta$ does $R$ have a non-trivial (real) eigenvector?
    2. Notice that $R$ corresponds to rotation by an angle $\theta$ about the origin; explain why your answer above makes sense geometrically.
  5. Consider the following matrices: $$A = \begin{bmatrix}2&1\\0&2\end{bmatrix} \hspace{2in} B = \begin{bmatrix}2&0\\0&2\end{bmatrix}.$$
    1. Find the characteristic polynomials of $A$ and $B$. (This requires very little work.)
    2. Find a non-trivial eigenvector for $A$, and two linearly independent eigenvectors for $B$.
    3. 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.
  6. Let $M$ be the following matrix: $$M = \begin{bmatrix}6&4&2\\-5&-3&-1\\-2&-2&-2\end{bmatrix}.$$
    1. Find the eigenvalues of $M$.
    2. Find a linearly independent set of three eigenvectors of $M$.
    3. Write down matrices $P$ and $D$ so that $D$ is diagonal, $P$ is invertible, and $M = PDP^{-1}$.
    4. Compute $M^3$.