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\pagestyle{fancy} \lhead{{\sf Homework 12 $|$ Due November 20 (Tuesday) }}
\rhead{\thepage} \cfoot{{\sf MthSc 311 $|$ Linear Algebra $|$ Fall 2012 
    $|$ M.~Macauley}}


\begin{document}

$\;$

\noindent\emph{Read}: Strang, Section 6.4, 8.3, 8.5, 10.1, 10.2. \\

\noindent\emph{Suggested short conceptual exercises}: Strang, Section
6.4, \#2, 7--10, 13--15, 18, 20--22, 25, 26. Section 8.3, \#1, 4, 6,
8--11, 17. Section 8.5, \#5, 10.


\begin{enumerate}

  % Strang Section 8.3, #3 (modified)
\item Consider the following three Markov matrices.
  \[
  \A=\begin{bmatrix}1&.2\\0&.8\end{bmatrix}\,\qquad\qquad
  \A=\begin{bmatrix}.2&1\\.8&0\end{bmatrix}\,\qquad\qquad
  \A=\begin{bmatrix}.5&.25&.25\\.25&.5&.25\\.25&.25&.5\end{bmatrix}\,.
  \]
  Carry out the following steps for each matrix.
  \begin{enumerate}
  \item Draw a weighted directed graph on $n$ (=$2$ or $3$) vertices
    showing the transition probabilities between states.
  \item Find the eigenvalues and steady-state eigenvector.
  \item Diagonalize $\A$ by writing $\A=\S\L\S^{-1}$ and compute
    $\ds\lim_{k\to\infty}\A^k$.
  \end{enumerate}
  


  % Strang Section 8.3, #5 (highly modified)
\item Every year, $10\%$ of young people become old, $0.1\%$ of young
  people become dead, and $5\%$ of old people become dead. Assume
  there are no births and no zombies.
  \begin{enumerate}
  \item Find the Markov matrix $\A$ that models this process:
    \[
    \begin{bmatrix}\text{young}\\\text{old}\\\text{dead}\end{bmatrix}_{k+1}
    =\begin{bmatrix}-&-&-\\-&-&-\\-&-&-\end{bmatrix}
    \begin{bmatrix}\text{young}\\\text{old}\\\text{dead}\end{bmatrix}_k.
    \]
  \item Find the steady-state eigenvector by inspection alone. 
  \item Find the eigenvalues and eigenvectors of $\A$. (Use a
    computer.)
  \item Suppose there are $1000$ young people initially. Compute the
    number of young, old, and dead people after $10$ years. (Again,
    use a computer, but also use $\A^k=\S\L^k\S^{-1}$.)
  \end{enumerate}



  % Complex Fourier series: Made up myself
\item Consider the set of all complex-valued $2\pi$-periodic
  functions, which means that $f(z+2\pi)=f(z)$ for all $z$. This is a
  vector space and the infinite set $\{e^{inx}:n\in\Z\}$ is a
  basis. Define an \emph{inner product} (i.e., dot product) on this
  space by
  \[
  \<f,g\>=\int_0^{2\pi}f(x)\bar{g(x)}\,dx.
  \]
  \begin{enumerate}
  \item Compute $\<e^{inx},e^{imx}\>$ and verify that this basis is
    indeed orthogonal. Recall that $\bar{e^{ix}}=e^{-ix}$, and be sure
    to consider the cases when $n=m$ and $n\neq m$ separately.
  \item Since $\{e^{inx}:n\in\Z\}$ is a basis, we can write any
    $2\pi$-periodic function $f(x)$ as
    \[
    f(x)=\;\;\cdots+c_{-2}e^{-2ix}+c_{-1}e^{-ix}+c_0+c_1e^{ix}+c_2e^{2ix}
    +\cdots\;\;=\sum_{n=-\infty}^\infty
    c_ne^{inx}\,.
    \]
    Derive a formula for $c_n$. [\emph{Hint}: Right-multiply both sides of
      the above equation by $\bar{e^{inx}}$ and integrate.]
  \item Give an \emph{orthonormal basis} for this vector space.
  \item Define the function $f(x)=e^x$ on the interval $[0,2\pi]$, and
    extend $f$ to be periodic. Sketch the graph of this function and
    compute its complex Fourier series. That is, compute the
    coefficients $c_n$. You'll need to compute $c_n$ (for $n\neq 0$)
    and $c_0$ separately.
  \end{enumerate}



  % 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}



  % 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 $\B$:
  $\bold{LU}$, $\bold{QR}$, $\bold{S\L S}^{-1}$, $\bold{Q\L Q}^T$?



  % Strang, Section 6.4, #24
\item What number $b$ in $\A=\begin{bmatrix}2&b\\1&0\end{bmatrix}$
  makes $\A=\bold{Q\L Q}^T$ possible? What number makes
  $\A=\bold{S\L\S}^{-1}$ impossible? What number makes $\A^{-1}$
  impossible?
  
\end{enumerate}


\end{document}



