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\pagestyle{fancy} \lhead{{\sf Homework 11 $|$ Due April 4 (Friday) }}
\rhead{\thepage} \cfoot{{\sf Math 311 $|$ Linear Algebra $|$ Spring 2014 
    $|$ M.~Macauley}}


\begin{document}

$\;$

\noindent\emph{Read}: Strang, Section 6.1, 6.2. \\

\noindent\emph{Suggested short conceptual exercises}: Strang, Section
6.1, \#1, 7, 8, 13, 16, 17, 19, 23, 25, 29. Section 6.2, \#3--5, 7,
11--14, 20--22, 24, 25, 29, 30, 32, 34.


\begin{enumerate}


  % Strang: Section 6.1, #2-4 (highly modified)
\item Consider the following matrices:
  \[
  \A=\begin{bmatrix}0&2\\1&1\end{bmatrix},\qquad\quad 
  \A^{2}=\begin{bmatrix}2&2\\1&3\end{bmatrix},\qquad\quad
  \A+\I=\begin{bmatrix}1&2\\1&2\end{bmatrix},\qquad\quad
  \A^{-1}=\begin{bmatrix}-1/2&1\\1/2&0\end{bmatrix}.
  \]
  \begin{enumerate}
  \item Find the eigenvalues and eigenvectors of $\A$.
  \item Diagonalize $\A$, by writing $\A=\S\L\S^{-1}$, where
    $\L$ is a diagonal matrix of eigenvalues and $\S$ is the matrix
    of corresponding eigenvectors. (No need to actually compute
    $\S^{-1}$ by taking the inverse; just put a $-1$ exponent on $\S$.)
  \item Repeat the previous parts for the matrices $\A^2$, $\A+\I$, and
    $\A^{-1}$.
  \end{enumerate}
  


  % Strang: Section 6.1, #6 (modified)
\item Find the eigenvalues of the following matrices:
  \[
  \A=\begin{bmatrix}1&0\\1&1\end{bmatrix},\qquad\qquad 
  \B=\begin{bmatrix}1&2\\0&1\end{bmatrix},\qquad\qquad 
  \A\B=\begin{bmatrix}1&2\\1&3\end{bmatrix},\qquad\qquad 
  \B\A=\begin{bmatrix}3&2\\1&1\end{bmatrix}.
  \]
  \begin{enumerate}
  \item Are the eigenvalues of $\A\B$ equal to the product of the
    eigenvalues of $\A$ and $\B$?
  \item Do $\A\B$ and $\B\A$ have the same eigenvalues?
  \end{enumerate}



  % Strang: Section 6.1, #9, 21 (modified)
\item Suppose $\lambda$ is an eigenvalue of $\A$. Prove the following:
  \begin{enumerate}
  \item $\lambda^2$ is an eigenvalue of $\A^2$.
  \item If $\A$ is invertible, then $\lambda^{-1}$ is an eigenvalue of
    $\A^{-1}$.
  \item $\lambda+c$ is an eigenvalue of $\A+c\I$, where $c$ is a
    constant.
  \item $\A$ and $\A^T$ have the same eigenvalues. 
  \end{enumerate}
  


  % Strang: Section 6.1, #16-17 (modified)
\item The \emph{characteristic polynomial} of $\A$ is
  $\chi_{\A}(\lambda)=\det(\A-\lambda\I)$. Suppose this factors (always
  possible) as
  \[
  \chi_{\A}(\lambda)=(\lambda_1-\lambda)(\lambda_2-\lambda)\cdots
  (\lambda_n-\lambda).
  \]
  \begin{enumerate}
    \item Plug in $\lambda=0$ and find a formula for $\det\A$ in terms
      of the eigenvalues of $\A$. 
    \item The \emph{trace} of $\A$, denoted $\tr\A$, is the sum of the
      diagonal entries which is also (amazingly!) equal to the sum of the
      eigenvalues. If $\A$ is $2\times 2$, then
      \[
      \A=\begin{bmatrix}a&b\\c&d\end{bmatrix}\mbox{\quad has\quad}
      \det(\A-\lambda\I)=\lambda^2-(a+d)\lambda+(ad-bc)=0.
      \]
      Write a formula for the chacteristic polynomial of a $2\times 2$
      matrix in terms of $\det\A$ and $\tr\A$. 
    \item Suppose $\A$ is an $n\times n$ matrix with characteristic
      polynomial $\chi_{\A}=\lambda^n+c_{n-1}\lambda^{n-1}+\cdots+
      c_1\lambda+c_0$.  Describe $\det\A$ and $\tr\A$ in terms of the
      $c_i$'s.
    \item Explain why $\A\B-\B\A=\I$ is impossible for $n\times n$ matrices.
  \end{enumerate}  




  % Made up myself
\item Find the eigenvalues and eigenvectors of $\A$, $\B$, and
  $\C$. Compute the traces and determinants of these matrices as well.
  \[
  \A=\begin{bmatrix}2&1&0\\0&1&0\\0&0&1\end{bmatrix},\qquad\qquad
  \B=\begin{bmatrix}1&1&0\\0&1&0\\0&0&1\end{bmatrix},\qquad\qquad
  \C=\begin{bmatrix}1&1&0\\0&1&1\\0&0&1\end{bmatrix}.
  \]



% Strang: Section 6.1, #14 (modified)
\item Recall that the following matrix rotates the $xy$-plane by the
  angle $\theta$:
  \[
  \Q=\begin{bmatrix}\cos\theta&-\sin\theta\\
  \sin\theta&\cos\theta\end{bmatrix}.
  \]
  \begin{enumerate}
  \item Solve $\det(\Q-\lambda\I)=0$ by the quadratic formula to find
    the eigenvalues of $\Q$.
  \item Find the eigenvectors of $\Q$ by solving
    $(\Q-\lambda\I)\x=\0$. Use $i^2=-1$.
  \end{enumerate}



  % Strang: Section 6.1, #32 (modified)
\item Suppose $\A$ has eigenvalues $0$, $3$, $5$ with linearly
  independent eigenvectors $\u$, $\v$, $\w$.
  \begin{enumerate}
  \item Give a basis for the nullspace and a basis for the column space. 
  \item Find a particular solution to $\A\x=\v+\w$. Find all solutions. 
  \item Explain why $\A\x=\u$ has no solution. 
  \end{enumerate}



  % Strang: Section 6.2, #2
\item If $\A$ has $\lambda_1=2$ with eigenvector $\x_1=\vv{1}{0}$ and
  $\lambda_2=5$ with $\x_2=\vv{1}{1}$, use $\S\L\S^{-1}$ to find
  $\A$. Note that no other matrix has the same $\lambda$'s and $\x$'s!

  
\end{enumerate}


\end{document}




