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\pagestyle{fancy} \lhead{{\sf Homework 6 $|$ Due September 28
    (Wednesday) }} \rhead{\thepage} \cfoot{{\sf Math 8530 $|$ Linear
    Algebra $|$ Fall 2016 $|$ 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 $A$. \smallskip
  \end{enumerate}


%%----------------------------------------------------------------------


\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. \smallskip
  \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
  $\R^2$. \smallskip


%%----------------------------------------------------------------------

%\item Consider the following matrices:
%  \[
%  A=\begin{bmatrix}-2&0\\0&-2\end{bmatrix}\,, \qquad\quad
%  B=\begin{bmatrix}0&-2\\-2&0\end{bmatrix}\,, \qquad\quad
%  C=\begin{bmatrix}3&1\\-1&1\end{bmatrix}\,.
%  \]
%  \begin{enumerate}
%  \item Determine the characteristic and minimal polynomials of $A$,
%    $B$, and $C$.
%  \item  Determine the eigenvectors and generalized eigenvectors
%    of $A$, $B$, and $C$. \\
%  \end{enumerate}

%%----------------------------------------------------------------------

\item Consider the following matrices:
  \[
  A=\begin{bmatrix} 2&-2&14\\0&3&-7\\0&0&2\end{bmatrix}\,,\qquad
  B=\begin{bmatrix} 0&-4&85\\1&4&-30\\0&0&3\end{bmatrix}\,,\qquad
  C=\begin{bmatrix} 2&2&1\\0&2&-1\\0&0&3\end{bmatrix}\,.
  \]
  A straightforward calculation shows that the characteristic polynomials are
  \[
  p_A(s)=p_B(s)=p_C(s)=(s-2)^2(s-3)\,.
  \]
  \begin{enumerate}
  \item Determine the eigenvectors and the minimal polynomials of each matrix
  \item Find a basis $\{v_1,v_2,v_3\}$ for $\R^3$ where $Bv_1=3v_1$,
    $Bv_2=2v_2$, and $(B-2I)v_3=v_2$. Write the matrix of this linear map
    with respect to this new basis. 
  \item Repeat the previous step for the matrix $C$. \smallskip
  \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. \smallskip
  \end{enumerate}

%%----------------------------------------------------------------------

\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}







  
