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\pagestyle{fancy} \lhead{{\sf Homework 7 $|$ Due March 6
    (Friday) }} \rhead{\thepage} \cfoot{{\sf Math 8530 $|$ Linear
    Algebra $|$ Spring 2015 $|$ M.~Macauley}}


\begin{document}

$\;$

\noindent\emph{Read}: Lax, Chapter 6, pages 58--69. \\

\begin{enumerate}

% Lax Exercise 5.10-11
\item Prove the following properties of the trace function:
  \begin{enumerate}
  \item $\tr AB=\tr BA$ for all $m\times n$ matrices $A$ and
    $n\times m$ matrices $B$.
  \item $\tr AA^T=\sum a_{ij}^2$ for all $n\times n$ matrices $A$. The
    quantity is the square of the \emph{Hilbert-Schmidt norm} of $A$. \\
  \end{enumerate}

  

\item Find the eigenvalues and corresponding eigenvectors for the
  following matrices over $\C$.
  \[
  (a)\quad\begin{bmatrix} 3&2 \\-2&3 \end{bmatrix}\hspace{15mm}
  (b)\quad\begin{bmatrix} 5&-6&-6\\-1&4&2\\3&-6&-4\end{bmatrix}\hspace{15mm}
  (c)\quad\begin{bmatrix} 3&1&-1\\2&2&-1\\2&2&0\end{bmatrix}.
  \]

\smallskip


\item
  \begin{enumerate}
  \item Show that if $A$ and $B$ are similar, then $A$ and $B$
    have the same eigenvalues. 
  \item Is the converse of Part~(a) true? Prove or disprove. \\
  \end{enumerate}



\item Let $A_\theta$ be a $3\times 3$ matrix representing a rotation
  of $\R^3$ through an angle $\theta$ about the $y$-axis.
  \begin{enumerate}
  \item Find the eigenvalues for $A_\theta$ over $\C$.
  \item Determine necessary and sufficient conditions on $\theta$ in
    order for $A_\theta$ to contain three linearly independent
    eigenvectors in $\R^3$. Justify your claim and interpret it
    geometrically. \\
  \end{enumerate}



\item Let $A$ be a $2\times 2$ matrix over $\R$ satisfying
  $A^T=A$. Prove that $A$ has 2 linearly independent eigenvectors in
  $\R^2$. \\



\item Let $A$ be an invertible $n\times n$ matrix. Prove that $A^{-1}$
  can be written as a polynomial in degree at most $n-1$. That is,
  prove that there are scalars $c_i$ such that
  \[
  A^{-1}=c_{n-1}A^{n-1}+c_{n-2}A^{n-2}+\cdots c_1A+c_0I\,.
  \] 

\smallskip


% Lax Exercise 6.2, modified
\item Let $A$ be an $n\times n$ matrix over $\C$ with distinct
  eigenvalues $\lambda_1,\dots,\lambda_n$. For a vector
  $z=(z_1,\dots,z_n)\in\C^n$, define the \emph{norm} of $z$ by
  \[
  ||z||=\Big(\sum_{i=1}^n|z_i|\Big)^{1/2}.
  \]
  \begin{enumerate}
  \item Prove that if $|\lambda_i|<1$ for all $i$, then $||A^Nz||\to
    0$ as $N\to\infty$ for all $z\in\C^n$.
  \item Prove that if $|\lambda_i|>1$ for all $i$, then
    $||A^Nz||\to\infty$ as $N\to\infty$ for all nonzero $z\in\C^n$.
  \end{enumerate}


\end{enumerate}

\end{document}







  
