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\pagestyle{fancy} \lhead{{\sf Homework 13 $|$ Due April 18 (Friday) }}
\rhead{\thepage} \cfoot{{\sf Math 3110 $|$ Linear Algebra $|$ Spring 2014 
    $|$ M.~Macauley}}


\begin{document}

$\;$

\noindent\emph{Read}: Strang, Section 6.4, 6.5. \\

\noindent\emph{Suggested short conceptual exercises}: Strang, 
Section 6.4, \#2, 7--10, 13--15, 18, 20--22, 25, 26. Section
6.5, \#1, 6, 10, 14--20, 27--31, 33. 


\begin{enumerate}


  % Strang, Section 6.4, #5-6 (highly modified)
\item Diagonalize the following matrices into $\Q\L\Q^T$, where
  $\Q$ is an \emph{orthogonal} matrix.
  \[
  \A=\begin{bmatrix}7&6\\6&-2\end{bmatrix}\,\qquad\qquad
  \B=\begin{bmatrix}0&-2&2\\-2&-1&0\\2&0&1\end{bmatrix}\,.
  \]
  Find \emph{all} orthogonal matrices that diagonalize $\A$. How many
  will diagonalize $\B$?



  % Strang, Section 6.4, #23
\item Which of these classes of matrices do $\A$ and $\B$ belong to:
  Invertible, orthogonal, projection, permutation, diagonalizable,
  Markov?
  \[
  \A=\begin{bmatrix}0&0&1\\0&1&0\\1&0&0\end{bmatrix}\qquad\qquad
  \B=\frac{1}{3}\begin{bmatrix}1&1&1\\1&1&1\\1&1&1\end{bmatrix}\,.
  \]
  Which of these factorizations are possible for $\A$ and $\B$:
  $\bold{LU}$, $\bold{QR}$, $\bold{S\L S}^{-1}$, $\bold{Q\L Q}^T$?



  % Hermitian matrices: Made up myself
\item Let $\A$ be an $n\times n$ \emph{Hermitian} matrix (also called
  \emph{self-adjoint}), which means that $\bar{\A}^T=\A$. Note that
  $\A$ may have complex-valued entries.
  \begin{enumerate}
  \item Which Hermitian matrices are also symmetric?
  \item Prove that all eigenvalues of $\A$ are real.
  \item Prove that if $\A\v=\lambda\v$ and $\A\w=\mu\w$ for
    $\lambda\neq\mu$, then $\v$ and $\w$ are orthogonal.
  \end{enumerate}
  These two proofs can be done by modifying the arguments in class
  that proved these statements for real symmetric matrices.



  % Part (d) is Strang, Section 6.4, #1 (modified), rest made up.
\item An real-valued $n\times n$ matrix is \emph{skew-symmetric} if
  $\A^T=-\A$.
  \begin{enumerate}
    \item What must be true about the diagonal entries of a
      skew-symmetric matrix? %Justify your answer.
    \item Prove that every non-zero eigenvalue of a skew-symmetric
      matrix is purely imaginary. [\emph{Hint}: Note that a
      complex eigenvalue $\lambda=a+bi$ is real if
      $\lambda=\bar\lambda$, whereas it is purely imaginary if
      $\lambda=-\bar\lambda$.]
    \item Prove that if $n$ is odd and $\A$ is skew-symmetric, then
      $\A$ must be singular. [\emph{Hint}: complex eigenvalues come in
      pairs, because $\A\v=\lambda\v$ implies
      $\bar{\A\v}=\A\bar{\v}=\bar{\lambda}\bar{\v}$.]
    \item Write $\A=\M+\N$, the sum of a \emph{symmetric} and a
      \emph{skew-symmetric} matrix:
      \[
      \A=\begin{bmatrix}1&2&5\\4&3&0\\7&6&5\end{bmatrix}=\M+\N\qquad\qquad
      (\M^T=\M,\;\;\N^T=-\N).
      \]
      \emph{Hint}: For any square matrix, $\M=\frac{1}{2}(\A+\A^T)$. What
      does this force $\N$ to be?
  \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:


\end{enumerate}


\end{document}

