\documentclass{beamer}
\usepackage{color,graphicx,psfrag}
\usepackage{beamerthemesplit}
\usetheme{Warsaw}
\usecolortheme{whale}
\title[$C$-Symmetric Matrices]{Towards a Classification of $3 \times 3$ $C$-Symmetric Matrices}
\author[Jay Daigle]{Jay Daigle\\ Advisor: Stephan Garcia
  \\\texttt{gjd02004@mymail.pomona.edu}
  \\\texttt{http://www.dci.pomona.edu/\textasciitilde jadagul}}
\institute{Pomona College}

%\date{June 15, 2007}
\ifx\pdfoutput\undefined
% we are running LaTeX, not pdflatex
\usepackage{graphicx}
\else
% we are running pdflatex, so convert .eps files to .pdf
\usepackage{graphicx}
\usepackage{epstopdf}
\fi
\newtheorem{thm}{Theorem}
\newtheorem{rmk}[thm]{Remark}
\newtheorem{prop}[thm]{Proposition}
\newtheorem{lem}[thm]{Lemma}
\newtheorem{dfn}[thm]{Definition}
\newtheorem{cor}[thm]{Corollary}
\newtheorem{conj}[thm]{Conjecture}
\newcommand{\del}{\Delta^S}
\newcommand{\delt}{\del(M)}
\newcommand{\newL}{\mathcal{L}^S}
\newcommand{\floor}[1]{\left\lfloor #1 \right\rfloor}
\newcommand{\ceil}[1]{\left\lceil #1 \right\rceil}

\definecolor{Blue}{rgb}{0,0,.8}
\definecolor{Green}{rgb}{0,.8,0}
\definecolor{Red}{rgb}{.8,0,0}

%------
\begin{document}
\frame{\maketitle}

\frame{
\begin{figure}
\includegraphics[scale=.4]{../pics/attenshun.jpg}
\end{figure}
}

\frame{\pause
\begin{figure}
\includegraphics[scale=.6]{../pics/funny-pictures-cat-couch-too-fast.jpg}
\end{figure}
}


\section{Definitions and Background}
\label{sec:background}


\frame{\frametitle{Motivating Examples}
\pause
\alert<2>{$$\left(  
\begin{array}{ccc}
    a&b&c\\
    b&d&e\\
    c&e&f\\
  \end{array} 
\right)$$}
\pause
$$\begin{array}{cc}
      \alert<3>{\left(
        \begin{array}{ccc}
          1&1&0\\
          0&1&1\\
          1&0&1\\
        \end{array}
      \right)}
  &
 \pause
      \alert<4>{\left(
        \begin{array}{ccc}
          \frac{1+\sqrt{3}i}{2}&0&0\\
          0&\frac{1-\sqrt{3}i}{2}&0\\
          0&0&2\\
        \end{array}
      \right)}
  \\
\end{array}$$
\vskip.1in
$$%\begin{array}{cc}
\pause
      \alert<5>{\left(
        \begin{array}{ccc}
          -5&0&4\\
          -4&-2&2\\
          1&4&-3\\
        \end{array}
      \right)}\pause
%&
 %     \alert<6>{\left(
  %      \begin{array}{ccc}
   %       1&0&1\\
    %      0&2&2\\
     %     0&0&0\\
      %  \end{array}
      %\right)}
%    \end{array}
$$    

\pause
\begin{block}
    {Fact}
Every matrix is similar to a complex symmetric matrix.
  \end{block}
}


\frame{\frametitle{Unitary Equivalence}
\pause
\begin{dfn}
Let $U$ be a $n \times n$ matrix.  If $U$ is an invertible isometry,
then we say $U$ is \alert<2>{unitary}.
\end{dfn}
\pause
\begin{dfn}
  If $A$ and $B$ are $n \times n$ matrices and $A=UBU^{-1}$ for some
  unitary matrix $U$, then $A$ is unitarily equivalent to $B$.
\end{dfn}
}

\frame{\frametitle{Outlines of the Problem} 
\pause
%  \begin{block}
    \begin{itemize}
    \item Every square matrix is similar to a complex symmetric matrix
      (CSM).\pause
    \item Not every square matrix is unitarily equivalent to a CSM
      (UECSM).\pause 
    \item Develop techniques to tell the difference.\pause
    \item Classify $3 \times 3$ UECSM.

    \end{itemize}
%  \end{block}
}

\frame{\frametitle{Why is this hard?}
\pause
Most useful invariants are similarity invariants.
\pause

 \begin{block}

 \begin{eqnarray*}
  \det(A)&=&\det(Q^{-1}BQ)\\
&=&\det(Q^{-1})\det(B)\det(Q)\\
&=&\det(Q^{-1})\det(Q)\det(B)\\
&=&\det(I)\det(B)=\det(B)\\
\end{eqnarray*}
 \end{block}
\pause

\begin{minipage}{.45\textwidth}
\begin{itemize}
\item Determinant
\item Trace
\item Eigenvalues
\end{itemize}
\end{minipage}
\begin{minipage}{.45\textwidth}
\begin{itemize}
\item Rank
\item Minimum Polynomial
\item Jordan Form
\end{itemize}
\end{minipage}
}

\section{Our Tools}
\frame{}
\frame{\frametitle{The Conjugate Transpose $T^*$}

\pause
  \begin{dfn}
    If $T=(a_{ij})$ is a square complex matrix, then we say that
    $T^*=(\overline{a_{ji}})$ is its \alert<2>{conjugate transpose}.
  \end{dfn}
\pause
\hfill
 
 \begin{minipage}[t]{.45\textwidth}
%    \begin{block}
      $$T=
      \left(
        \begin{array}{ccc}
          1&6&i\\
          2i&-3+4i&4-3i\\
          -2&5&3
        \end{array}
      \right)
      $$
%    \end{block}
  \end{minipage}
 \pause
  \hfill
  \begin{minipage}[t]{.45\textwidth}
%    \begin{block}
      $$T^*=
      \left(
        \begin{array}{ccc}
          1&-2i&-2\\
          6&-3-4i&5\\
          -i&4+3i&3
        \end{array}
      \right)
      $$
%    \end{block}        


  \end{minipage}
  \hfill
}

\frame{\frametitle{Conjugations}
\pause
\begin{dfn}
A conjugation $C$ is a isometric antilinear involution.
\end{dfn}
\pause

\begin{itemize}
\item Isometric: leaves sizes and angles unchanged. \pause
\item Antilinear: $C(\lambda x) = \overline{\lambda}Cx.$ \pause
\item Involution: $C \circ C = I.$
\end{itemize}
\pause

\begin{dfn}
  The \alert<5>{standard conjugation} $J$ takes a vector to its conjugate:
$$J(x_1,x_2,\dots,x_n)=(\overline{x_1},\overline{x_2},\dots,\overline{x_n}).$$
\end{dfn}
}

\frame{\frametitle{$C$-symmetry}
\pause
  \begin{dfn}
    We say a matrix $T$ is \alert<2>{$C$-symmetric} if there exists a conjugation
    $C$ such that $T=CT^*C$.
  \end{dfn}


\pause
\begin{thm}
    A matrix is UECSM if and only if it is
    $C$-symmetric for some conjugation $C$.
\end{thm}

}

\frame{\frametitle{Derivative Results}
\pause
  \begin{itemize}
  \item Every $2 \times 2$ matrix is UECSM.\pause
    % \item partial isometries?
  \item Rank 1 matrices are  UECSM.\pause
  \item Direct sum of UECSM is UECSM.
  \end{itemize}
}


\frame{\frametitle{A Brief Review of Eigenvectors}
\pause
  \begin{dfn}
    Let $T$ be a matrix.  Then if there exists a vector $v$ and a
    scalar $\lambda$ such that $Tv-\lambda v=0$, then we say that
    $\lambda$ is an eigenvalue of $T$ and $v$ is an eigenvector with
    eigenvalue $\lambda$.
  \end{dfn}
\pause
  \begin{dfn}
    Let $T,v,\lambda$ be as above.  If there exists a natural number
    $n$ such that $(T-\lambda I)^n v =0$ then $v$ is a generalized
    eigenvector of $T$ with eigenvalue $\lambda$.
  \end{dfn}
}

\frame{\frametitle{Conjugating Eigenvectors}
\pause
  \begin{block}{Fact}
    $T$ and $T^*$ have conjugate eigenvalues.
  \end{block}
\pause
$$Tv=\lambda v \Rightarrow T^*u = \overline{\lambda}u.$$
\pause

  \begin{block}
  
  $$\lambda v = T v = C T^* C v$$
$$\overline{\lambda}(C v)=C(\lambda v) = CT v = T^* (Cv)$$
  
Thus $Cv$ is an eigenvector of $T^*$ with eigenvalue $\overline{\lambda}.$
 \end{block}
\pause
Thus $C$ must take eigenvectors of $T$ to corresponding eigenvectors
of $T^*.$ 



}
\section{Next Steps}
\label{sec:nextsteps}

\subsection{Classification of $3 \times 3$ Matrices}
\label{sec:3by3}
\frame{\frametitle{How to Classify?}
\pause
  \begin{itemize}
    \item Unitary Equivalence is an equivalence relation.\pause
    \item Which representative should we use?\pause
  \end{itemize}

  \begin{block}{Schur's Theorem}
    Every square matrix is unitarily equivalent to an upper triangular matrix.
  \end{block}\pause

      $$
      \left(
        \begin{array}{ccc}
          \lambda_1&a&b\\
          0&\lambda_2&c\\
          0&0&\lambda_3
        \end{array}
      \right)
      $$

}



\frame{\frametitle{A (Pseudo)-General Algorithm}
\pause

$$T=\left(
  \begin{array}{ccc}
    0&a&b\\
    0&0&0\\
    0&0&1
  \end{array}
\right)
$$
\pause
$$u_0=
\left(\begin{array}{c}
  1\\0\\0
\end{array}\right), 
u_1=\left(\begin{array}{c}
  b\\0\\1
\end{array}\right),
v_0=
\left(\begin{array}{c}
0\\1\\0
 % \tfrac{\overline{\lambda}}{\overline{ac-b}}\\ \tfrac{\overline{a\lambda}}{\overline{b-ac}}\\1
\end{array}\right),
v_1=
\left(\begin{array}{c}
0\\0\\1
  %0\\\tfrac{\overline{\lambda}-1}{\overline{c}}\\1
\end{array}\right)$$
\pause
$|b+0+0|=|0+0+0|$

\pause

$u_0 \to v_0,~ u_1 \to v_1,~ u_{\lambda} \to v_{\lambda}.$


}



\frame{\frametitle{Our Cases}
\pause


$$
\begin{array}{ccc}
 \textcolor{Green}{ \left(
  \begin{array}{ccc}
    0&a&b\\
    0&0&c\\
    0&0&0
  \end{array}
\right)}\pause
&
\textcolor{Blue}{\left(
  \begin{array}{ccc}
    0&a&b\\
    0&0&c\\
    0&0&1
  \end{array}
\right)}
&
\textcolor{Blue}{\left(
  \begin{array}{ccc}
    0&a&b\\
    0&1&c\\
    0&0&\lambda
  \end{array}
\right)}
\end{array}
$$


}

\frame{\frametitle{One Eigenvalue}
$$\left(
  \begin{array}{ccc}
    0&a&b\\
    0&0&c\\
    0&0&0
  \end{array}
\right)
$$
\pause
$$
\begin{array}{ccc} \begin{array}{c}
 {\textcolor{Green}{ \left(
  \begin{array}{ccc}
    0&0&b\\
    0&0&c\\
    0&0&0\\
  \end{array}
\right)}}\\
\mbox{Rank 1}
\end{array}
\pause
&
\begin{array}{c} {\textcolor{Green}{\left(
  \begin{array}{ccc}
    0&a&b\\
    0&0&0\\
    0&0&0\\
  \end{array}
\right)}}\\
\mbox{Rank 1}
\end{array}
\pause
&
\begin{array}{c}
{\textcolor{Blue}{\left(
  \begin{array}{ccc}
    0&a&b\\
    0&0&c\\
    0&0&0\\
  \end{array}
\right)}}\\
|a|=|c|
\end{array}
\end{array}
$$
}

\frame{\frametitle{ Two Eigenvalues}

$${\left(
  \begin{array}{ccc}
    0&a&b\\
    0&0&c\\
    0&0&1
  \end{array}
\right)}
$$
\pause

$$
\begin{array}{ccc}
  \begin{array}{c}
\textcolor{Green}{\left(
  \begin{array}{ccc}
    0&0&b\\
    0&0&c\\
    0&0&1\\
  \end{array}
\right)}\\
\mbox{Rank 1}
\end{array}
&
\begin{array}{c}
\textcolor{Green}{\left(
  \begin{array}{cc|c}
    0&a&0\\
    0&0&0\\
    \hline
    0&0&1\\
  \end{array}
\right)}\\
2 \times 2 \oplus 1 \times 1
\end{array}
\pause
&
\begin{array}{c}
\textcolor{Red}{\left(
  \begin{array}{ccc}
    0&a&b\\
    0&0&0\\
    0&0&1\\
  \end{array}
\right)}
\\
\mbox{Angle Test}
\end{array}
\end{array}
$$\pause

$$
\begin{array}{cc}
  

\left(
  \begin{array}{ccc}
    0&a&b\\
    0&0&c\\
    0&0&1\\
  \end{array}
\right)
&
\left(
  \begin{array}{ccc}
    0&a&0\\
    0&0&c\\
    0&0&1\\
  \end{array}
\right)

\end{array}$$
}

\frame{\frametitle{Three Eigenvalues}
$$\left(
  \begin{array}{ccc}
    0&a&b\\
    0&1&c\\
    0&0&\lambda
  \end{array}
\right)
$$\pause


$$
\begin{array}{ccc}
 \begin{array}{c} 
{ \textcolor{Green}{\left(
  \begin{array}{c|cc}
    0&0&0\\
\hline
    0&1&c\\
    0&0&\lambda\\
  \end{array}
\right)}}\\
2 \times 2
\end{array}
&
\begin{array}{c}
{\textcolor{Green}{\left(
  \begin{array}{cc|c}
    0&a&0\\
    0&1&0\\
\hline
    0&0&\lambda\\
  \end{array}
\right)}}\\
2 \times 2 \oplus 1 \times 1\
\end{array}
&
\begin{array}{c}
{\textcolor{Green}{\left(
  \begin{array}{ccc}
    0&0&b\\
    0&1&0\\
    0&0&\lambda\\
  \end{array}
\right)}}\\

2 \times 2 \oplus 1 \times 1
\end{array}
\end{array}
$$\pause
$$
\begin{array}{cc}
  
\begin{array}{c}
{\textcolor{Red}{\left(
  \begin{array}{ccc}
    0&a&b\\
    0&1&0\\
    0&0&\lambda\\
  \end{array}
\right)}}\\
\mbox{Angle Test}
\end{array}
&
\begin{array}{c}
{\textcolor{Red}{\left(
  \begin{array}{ccc}
    0&0&b\\
    0&1&c\\
    0&0&\lambda\\
  \end{array}
\right)}}\\
\mbox{Angle Test}
\end{array}
\end{array}
$$\pause




}

\frame{\frametitle{Partial Isometries}

\begin{dfn}
A matrix $T$ is a partial isometry if there exists a unitary matrix
$U$ and an orthogonal projection $P$ such that $T=UP.$
\end{dfn}
\pause
\begin{prop}
  Every $3 \times 3$ partial isometry is UECSM.
\end{prop}
\pause
\begin{conj}
  Every $3 \times 3$ UECSM is a rank 1 matrix, a $2 \times 2 \oplus 1
  \times 1$, or some multiple of a partial isometry, plus some
  multiple of the identity matrix.
\end{conj}

}

\frame{
\begin{figure}
\includegraphics[scale=.3]{../pics/funny-pictures-anti-gravity-cat-chalkboard.jpg}
\end{figure}
}






\subsection{}
\frame{}
\frame{\tiny{$\langle2*u111*u211+2*u112*u212+2*u121*u221+2*u122*u222+2*u131*u231+2*u132*u232,2*u111*u311+2*u112*u312+2*u121*u321+2*u122*u322+2*u131*u331+2*u132*u332 ,2*u211*u311+2*u212*u312+2*u221*u321+2*u222*u322+2*u231*u331+2*u232*u332,2*u111*u212+2*u131*u232-2*u122*u221+2*u121*u222-2*u112*u211-2*u132*u231,2*u121*u322-2*u112*u311-2*u132*u331+2*u111*u312+2*u131*u332-2*u122*u321,-2*u222*u321+2*u221*u322-2*u212*u311-2*u232*u331+2*u211*u312+2*u231*u332,1-u111^2-u112^2-u121^2-u122^2-u131^2-u132^2,1-u211^2-u212^2-u221^2-u222^2-u231^2-u232^2,1-u311^2-u312^2-u321^2-u322^2-u331^2-u332^2,-2*s111*u111+2*s112*u112-2*s121*u211+2*s122*u212-2*s131*u311+2*s132*u312,2*u111*a1-2*s111*u121-2*u112*a2-2*s121*u221+2*s112*u122+2*s122*u222
-2*s131*u321+2*s132*u322+2*u121,2*u111*b1+2*u121*c1-2*u112*b2-2*u122*c2+2*u131*q1-2*u132*q2
-2*s111*u131-2*s121*u231+2*s112*u132+2*s122*u232-2*s131*u331+2*s132*u332,-2*s121*u111+2*s122*u112-2*s221*u211+2*s222*u212-2*s231*u311+2*s232*u312,-2*u212*a2+2*u211*a1+2*s222*u222-2*s221*u221+2*s122*u122
-2*s121*u121+2*s232*u322-2*s231*u321+2*u221,2*u211*b1-2*u212*b2-2*s121*u131-2*u232*q2+2*u231*q1-2*u222*c2+2*u221*c1
-2*s231*u331+2*s222*u232-2*s221*u231+2*s122*u132+2*s232*u332,-2*s131*u111+2*s132*u112-2*s231*u211+2*s232*u212-2*s331*u311+2*s332*u312,2*s132*u122-2*s131*u121-2*s231*u221+2*s232*u222-2*s331*u321
+2*s332*u322+2*u321+2*u311*a1-2*u312*a2,-2*u322*c2-2*u332*q2+2*u331*q1-2*s131*u131+2*s232*u232-2*s231*u231
+2*s132*u132-2*s331*u331+2*s332*u332+2*u321*c1+2*u311*b1-2*u312*b2,2*s112*u111+2*s122*u211+2*s131*u312+2*s121*u212+2*s111*u112+2*s132*u311,-2*u122-2*u111*a2+2*s111*u122-2*u112*a1+2*s131*u322+2*s132*u321
+2*s112*u121+2*s122*u221+2*s121*u222,-2*u132*q1-2*u131*q2-2*u111*b2-2*u112*b1-2*u121*c2-2*u122*c1
+2*s111*u132+2*s112*u131+2*s121*u232+2*s122*u231+2*s131*u332+2*s132*u331,2*s222*u211+2*s221*u212+2*s122*u111+2*s121*u112+2*s231*u312+2*s232*u311,2*s231*u322-2*u222+2*s222*u221-2*u211*a2-2*u212*a1+2*s121*u122
+2*s122*u121+2*s221*u222+2*s232*u321,-2*u231*q2-2*u211*b2-2*u221*c2+2*s221*u232-2*u232*q1-2*u222*c1+2*s231*u332
+2*s222*u231-2*u212*b1+2*s121*u132+2*s122*u131+2*s232*u331,2*s131*u112+2*s331*u312+2*s232*u211+2*s132*u111+2*s231*u212+2*s332*u311,-2*u322-2*u311*a2-2*u312*a1+2*s131*u122+2*s132*u121+2*s231*u222
+2*s232*u221+2*s331*u322+2*s332*u321,-2*u322*c1-2*u321*c2-2*u331*q2+2*s231*u232+2*s232*u231+2*s331*u332
+2*s332*u331-2*u312*b1-2*u332*q1+2*s131*u132+2*s132*u131-2*u311*b2\rangle$}

}
\frame{
\begin{figure}
\includegraphics[scale=.3]{../pics/funny-pictures-cat-waits-for-bird}
\end{figure}
}


\frame{
\pause
\begin{block}{To Paraphrase Richard Feynman:}
  Math is like sex.  Sure, it may give some practical results, but
  that's not why we do it.
\end{block}
}
\end{document}