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\pagestyle{fancy} \lhead{{\sf Homework 13 $|$ Due November 30 (Friday) }}
\rhead{\thepage} \cfoot{{\sf MthSc 311 $|$ Linear Algebra $|$ Fall 2012 
    $|$ M.~Macauley}}


\begin{document}

$\;$

\noindent\emph{Read}: Strang, Section 6.5, 6.6, 10.1. \\

\noindent\emph{Suggested short conceptual exercises}: Strang, Section
6.5, \#1, 6, 10, 14--20, 27--31, 33. Section 6.6, \#1, 4, 7,
8, 12--15, 17, 20.



\begin{enumerate}

% Strang Section 10.1, #14-15 (modified almost beyond recognition)
\item Consider the following permutation matrix:
  \[
  \P_4=\begin{bmatrix}0&0&0&1\\1&0&0&0\\0&1&0&0\\0&0&1&0\end{bmatrix}
  \]
\begin{enumerate}
  \item Find the four (complex-valued) eigenvalues and eigenvectors of
    $\P_4$.
  \item Write out the four eigenvalues you just found in polar form,
    $\lambda=Re^{i\theta}$. Write the corresponding eigenvectors in
    polar form, normalized so the first entry is 1, i.e.,
    $\v_k=(e^{i0},-,-,-)=(1,-,-,-)$.
  \item Without actually computing them, venture a guess as to what the
    eigenvalues and eigenvectors of $\P_6$ are -- the $6\times 6$
    matrix with five $1$'s below the diagonal and one up the
    upper-right corner.
\end{enumerate}



% Made up myself.
\item Consider the following two quadratic forms:
  \[
  f(x_1,x_2)=2x_1^2-4x_1x_2+5x_2^2,\qquad\qquad
  f(x_1,x_2)=2x_1^2+6x_1x_2+2x_2^2.
  \]
  Carry out the following steps for each of these functions.
  \begin{enumerate}
  \item Write $f(x_1,x_2)=\x^T\A\x$ for some $\A$.
  \item Diagonalize $\A$ by writing $\A=\Q\L\Q^T$ for an orthogonal
    matrix $\Q$ that additionally has $\det\Q=+1$.  [\emph{Hint}:
      Every $2\times 2$ orthogonal matrix with determinant $1$ is a
      \emph{rotation matrix}
      $\Q=\begin{bmatrix}\cos\theta&-\sin\theta\\ 
      \sin\theta&\cos\theta\end{bmatrix}$
      for some $0\leq\theta<2\pi$. Find $\theta$.]
  \item Introduce new coordinates by setting $\x=\Q\y$, and substitute
    $\y=\Q^T\x$ into the quadratic form $\x^T\A\x$. Write the result
    both as an equation $f(y_1,y_2)$, and in matrix form, $\y^T\L\y$.
  \item Sketch the conic section $\x^T\A\x=1$ on the $x_1x_2$-plane,
    and sketch $\y^T\L\y=1$ on the $y_1y_2$-plane (okay to use a computer).
  \item Write $f(x_1,x_2)$ as either the sum or the difference of two
    squares. (Only one will be possible.)
  \end{enumerate}



% Strang Section 6.5, #9 (modified)
\item Consider the following quadratic form:
  \[
  \x^T\A\x=\begin{bmatrix}x_1&x_2&x_3\end{bmatrix}
  \begin{bmatrix}&&&&\\&&\A&&\\&&&&\end{bmatrix}
  \vvv{x_1}{x_2}{x_3}=4(x_1-x_2+2x_3)^2.
  \]
  Find the $3\times 3$ matrix $\A$ and its pivots, rank,
  eigenvalues, and three (upper-left) subdeterminants:



  % 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.
\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 $\B_i=\M^{-1}\A_i\M$.
  \[
  \A_1=\begin{bmatrix}1&0\\1&0\end{bmatrix}\!,\;
  \B_1=\begin{bmatrix}1&0\\0&0\end{bmatrix}\!,\; 
  \A_2=\begin{bmatrix}1&1\\1&1\end{bmatrix}\!,\; 
  \B_2=\begin{bmatrix}1&-1\\-1&1\end{bmatrix}\!,\; 
  \A_3=\begin{bmatrix}1&2\\3&4\end{bmatrix}\!,\; 
  \B_3=\begin{bmatrix}4&3\\2&1\end{bmatrix}\!.
  \]



% 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.



% 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.    
\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 $\J_i^{-1}$

\end{enumerate}


\end{document}



