\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}