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\pagestyle{fancy} \lhead{{\sf Homework 14 $|$ Due April 25 (Friday) }}
\rhead{\thepage} \cfoot{{\sf Math 3110 $|$ Linear Algebra $|$ Spring 2014 
    $|$ M.~Macauley}}


\begin{document}

$\;$

\noindent\emph{Read}: Strang, Section 6.6, 6.7. \\

\noindent\emph{Suggested short conceptual exercises}: Strang, Section
6.6, \#1, 4, 7, 8, 12--15, 17, 20. Strang, Section 6.7, \#3, 9--13,
15.



\begin{enumerate}


  % Made up myself
\item Suppose $\B=\M^{-1}\A\M$ is an $n\times n$ matrix. The
  relationship between $\A$, $\B$, and $\M$ as functions $\R^n\to\R^n$
  is shown in the following \emph{commutative diagram}:
  \[
  \xymatrix{\R^n\ar[r]^B\ar[d]_M&\R^n\ar[d]^M\\\R^n\ar[r]^A&\R^n}
  \]
  Remember that matrix multiplication represents function composition,
  and so should be read from \emph{right-to-left}.
  \begin{enumerate}
  \item Draw a commutative diagram showing
    $\B^2=(\M^{-1}\A\M)(\M^{-1}\A\M)=\M^{-1}\A^2\M$. [\emph{Hint}:
    Imagine ``stacking'' two diagrams horizontally.]
  \item Suppose that $\B=\M^{-1}\A\M$ and $\C=\N^{-1}\B\N$. Draw a
    commutative diagram showing how $\A$ is similar to $\C$. Write
    this out algebraically as well. You have just proven that
    similarity is \emph{transitive}.
  \item Suppose $\A$ and $\B$ have the same eigenvalues
    $\lambda_1,\dots,\lambda_n$, all distinct. Prove that $\A$ and
    $\B$ are similar.
  \item Show by example how the result in Part~(c) fails if the
    eigenvalues are not distinct. \medskip
\end{enumerate}


  
  % Strang Section 6.6, #3 (modified)
\item Show that each pair $\A_i$ and $\B_i$  are similar by
  finding $\M_i$ such that $\B_i=\M_i^{-1}\A_i\M_i$.
%For each pair $\A_i$ and $\B_i$, show that they are similar by
%  finding $\M_i$ such that $\B_i=\M^{-1}\A_i\M$.
  \[
  \A_1=\begin{bmatrix}1&0\\1&0\end{bmatrix}\!,\;
  \B_1=\begin{bmatrix}1&0\\0&0\end{bmatrix}\!,\; 
  \A_2=\begin{bmatrix}1&1\\1&1\end{bmatrix}\!,\; 
  \B_2=\begin{bmatrix}1&-1\\-1&1\end{bmatrix}\!,\; 
  \A_3=\begin{bmatrix}1&2\\3&4\end{bmatrix}\!,\; 
  \B_3=\begin{bmatrix}4&3\\2&1\end{bmatrix}\!.
  \] 
  \vspace{0mm}



% Strange Section 6.6, #6 (modified)
\item There are sixteen $2\times 2$ matrices whose entries are $0$'s
  and $1$'s. Partition these matrices into ``families'' (equivalence
  classes) where similar matrices go into the same family. \bigskip



% Strang Section 6.6, #18 (modified almost beyond recognition)
\item Let $\A$ and $\B$ be $n\times n$ matrices with $\B$ invertible. 
  \begin{enumerate}
    \item Prove that $\A\B$ is similar to $\B\A$.
    \item Illustrate your proof from Part~(a) by correctly labeling
      the six maps in the following commutative diagram:
      \[
      \xymatrix{\R^n\ar[r]\ar[d]&\R^n\ar[r]&\R^n\ar[d]&\\
        \R^n\ar[r]&\R^n\ar[r]&\R^n}
      \]
    \item Conclude that $\A\B$ and $\B\A$ have the same eigenvalues. \medskip
\end{enumerate}



% Strang Section 6.6, #10 (highly modified)
\item Consider the following Jordan blocks with eigenvalue $\lambda$:
  \[
  \J_2=\begin{bmatrix}\lambda&1\\0&\lambda\end{bmatrix},\qquad
  \J_3=\begin{bmatrix}\lambda&1&0\\0&\lambda&1\\0&0&\lambda\end{bmatrix},\qquad
  \J_4=\begin{bmatrix}\lambda&1&0&0\\0&\lambda&1&0\\0&0&\lambda&1\\
  0&0&0&\lambda\end{bmatrix}.
  \]
  For each $\J_i$, compute $\J_i^2$ and $\J_i^3$. Guess the form of
  $\J_i^k$. Set $k=0$ to find $\J_i^0$ and $k=-1$ to find $\J_i^{-1}$. \medskip



% Made up myself
\item Suppose $\A$ is a $4\times 4$ matrix that has exactly two
  distinct eigenvalues, $\lambda=0$ and $\lambda=2$, but you do
  not know how many of each occurs.
  \begin{enumerate}
  \item Write down a list of matrices such that $\A$ must is similar
    to exactly one matrix on your list. 
  \item For each matrix above, find the number of linearly independent
    eigenvectors for $\lambda=0$ and for $\lambda=2$. Recall that this is
    the dimension of $N(A-\lambda I)$. \medskip
  \end{enumerate}



% Strang Section 6.7, #1-2 (modified)
\item Consider the following matrices:
  $\A=\begin{bmatrix}1&3\\2&6\end{bmatrix},\quad
  \A^T\A=\begin{bmatrix}5&15\\15&45\end{bmatrix},\quad
  \A\A^T=\begin{bmatrix}10&20\\20&40\end{bmatrix}$.
\begin{enumerate}
  \item Find the eigenvalues $\sigma_1^2$, $\sigma_2^2$ and unit
    eigenvectors $\v_1$, $\v_2$ of $\A^T\A$. 
  \item For the $\sigma_i\neq 0$, compute $\u_i=\A\v_i/\sigma_i$ and
    verify that indeed $||\u_i||=1$. Find the other $\u_i$ by computing
    the other unit eigenvector of $\A\A^T$.
  \item Write out the singular value decomposition (SVD),
    $\A=\U\bold{\Sigma}\V^T$.
  \item Write down orthonormal bases for the four fundamental
    subspaces of $\A$.
  \item Describe \emph{all} matrices that have the same four fundamental
    subspaces.
\end{enumerate}

\end{enumerate}


\end{document}



