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\def\borCo{L}
\def\SID{{Symbolic Inequality Domain}}
\def\POD{\succeq}
\def\upA{\stackrel{\rightarrow} }
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\def\cC{{\cal C}}
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\def\uM{{\upA M}}
\def\uZ{{\color{blue}{\upA \miZ}}}


\def\ux{{\color{red}{\upA \unkx}}}
\def\uX{{\color{red}{\upA \unkX}}}
\def\unkx{{\color{red}{x}}}
\newcommand{\unky}{{\textcolor{red}{y}}}
\newcommand{\unkX}{{\textcolor{red}{X}}}
   \def\unkxj{{\color{red}{x_j}}}
\def\unkxone{{\color{red}{x_1}}}
\def\unkxtwo{{\color{red}{x_2}}}
\def\unkxthree{{\color{red}{x_3}}}
\def\unkxm{{\color{red}{x_m}}}

\def\XXj{{\color{red}{X_j}}}
\def\unkXone{{\color{red}{X_1}}}
\def\unkXtwo{{\color{red}{X_2}}}
\def\unkXm{{\color{red}{X_m}}}
\def\unkT{{\color{red}{T}}}


\def\uMtil{{\upA{\tilde{M}}}}
\def\bLM{{\stackrel{\rightarrow}\borCo(\upA M)}}
\def\bL{{\stackrel{\rightarrow}\borCo}}
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\newcommand{\Q}{{\cal Q}}
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\newtheorem{thm}{Theorem}

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\newcommand{\CMxIs}{{\textcolor{red}{CONVEX MATRIX INEQUALITIES}}}
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\newcommand{\efZj}{\textcolor{blue}{\ESF{Z}_j}}

%------------------
 % mcc vinni macros

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\newcommand{\NDnfor}[1]{{\mathcal N}{\mathcal F}(n)_{#1,\text{for}}}
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\newcommand{\fgg}{\chi}

\newcommand{\dr}{}
%\newcommand{\dr}{{ \mathbb  }}
\newcommand{\erf}{\tt}

\def\smatn{\bbS \bbR^{n \times n}  } % $\smatn$

\def\gtupn{(\smatn)^g}

\def\bbS{{\mathbb S}}
\def\gtup{\bbS^g}


\def\cF{\mathcal F}
\def\cA{\mathcal A}



\def\nxn{{n \times n}}

%------------------------

\def\DEF{ \bf}
\def\cCp{ {\cal C}_p}

\def\demi{\frac{1}{2} }

\def\bbR{ {\bf R}}
\def\bR2{ {{\bf R}^2} }
\def\mRrr{ $\bbR^{r \times r}$ }
\def\mSRrr{ $ S \bbR^{r \times r}$ }


\def\efupdH{{\color{MyGreen}{\upA {H} }}}
\def\uAll{{A_1, A_2 ,\unkX, \unkY}}
\def\uAllX{{A_1, A_2 ,\unkXone, \unkXtwo}}

\def\unkcL{\color{blue}{\cal L} }
\def\udelta{\color{orange}{\delta}}

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\date{December 2000}



\def\smthm{ {\tiny\color{red} THM \color{black} }}

\begin{document}
\slidepagestyle{fancy}

\bfseries

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\Header{ ONE $R^{2 \times 2}$ TEST OF CONVEXITY}
\begin{slide}

%{\small{
{\ }
\vspace{-.6cm}
$$p(\unkx): =\unkx^4 + (\unkx^4)^T$$
  \begin{equation*}
\demi
p(\efX) +
\demi
p({\efY)} -
p(\demi \efX +
\demi
{ \efY} )
\succeq 0 ? \quad \ {\mbox{Try matrices}}
\end{equation*}

  $
   \efX=
  \left(\begin{array}{cc}
    1 & 1 \\
    0 & 1 \
  \end{array}
  \right)
  ,
   \efY=
  \left(\begin{array}{cc}
    1 & 3 \\
    2 & -1 \
  \end{array}
  \right),
$
$
\demi \efX + \demi \efY=
  \left(\begin{array}{cc}
    1 & 2 \\
    1 & 0 \
  \end{array}
  \right)
$%}}
{\tiny{
$$
\demi p( \efX  ) =
\demi \left[
  \left(\begin{array}{cc}
     1 & 1 \\
    0 & 1 \
  \end{array}
  \right)^4    + \left(\begin{array}{cc}
    1 & 0 \\
    1 & 1 \
  \end{array}
  \right)^4
 \right]
  =
   \left(\begin{array}{cc}
    1 & 2 \\
    2 & 1 \
  \end{array}
  \right)
$$
$
\demi p( \efY ) =
  \left(\begin{array}{cc}
    49 & 0 \\
     0 & 49 \
  \end{array}
  \right)
$
  and
$
p(\demi \efX + \demi \efY ) =
  \left(\begin{array}{cc}
    22 & 15 \\
    15 & 12 \
  \end{array}
  \right)
$.
  }}
Then
$$
  \left(\begin{array}{cc}
    1 & 2 \\
    2 & 1 \
  \end{array}
  \right)
   +
   \left(\begin{array}{cc}
    49 & 0 \\
     0 & 49 \
  \end{array}
  \right)
  -
  \left(\begin{array}{cc}
    22 & 15 \\
    15 & 12 \
  \end{array}
  \right)
     \succeq 0 \quad\quad {\mbox{i.e., is PSD}}
    $$

%   {\tiny{
%That  is, it is  PsD (Pos Semi Def).}}%}}
%\begin{equation*}
%\demi p(\uX) + \demi p({\efuY)} -
%p(\demi \uX + \demi { \efuY} ) \succeq 0
%\end{equation*}

\end{slide}

%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{slide}
    %\addtocounter{slide}{-1}
    \slidepagestyle{empty}

\centerline{\large%\bfseries
%\color{blue}Manipulating Matrix Inequalities Automatically}
{ \color{blue} Convex Matrix Inequalities vs Linear MIs} }
\bigskip
%\bigskip

\centerline{Scott McCullough \quad \quad University of Florida }

%\medskip
\centerline{Victor Vinnikov\quad \quad Ben Gurion University}

%\medskip
\centerline{Mihai Putinar \quad \quad Ben Gurion University}

%\medskip
\centerline{Nick Slinglend \quad \quad UC San Diego}


\medskip
\centerline{\bfseries Camino, de Oliveira,  Shopple, Skelton \quad
 UCSD}


\bigskip {\small
\centerline{I am Bill Helton \quad \quad  UCSD}
%\bigskip
%\centerline{ \color{green}
%{\small{ Grad Student Programming: John Shopple, Lim}}}
 %\color{black}
\bigskip
}

\centerline{
\color{red}{ NCAlgebra}\footnote{
\color{red}{
Helton, Stankus (CalPoly SanLObispo),
Miller (Archemedies Inc)}}
\color{black} }



%\begin{center}\small
%   Department of Mathematics \\
%   Department of Mechanical and Aerospace Engineering
%\end{center}


\end{slide}




\def\uH{{\color{green}{\upA \updH}}}
\newcommand{\updH}{{\textcolor{green}{H}}}
\newcommand{\updh}{{\textcolor{green}{h}}}
%%%

%\def\uH{{\color{MyGreen}{\upA \updH}}}
%\newcommand{\updH}{{\textcolor{MyGreen}{H}}}
%\newcommand{\updh}{{\textcolor{MyGreen}{h}}}
\newcommand{\updHH}{{\textcolor{MyGreen}{H_1}}}
\newcommand{\updK}{{\textcolor{MyGreen}{H_2}}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\dh{\color{blue} h \color{black}}
\def\dH{\color{blue} H \color{black}}



%\def\dh{\color{green} h \color{black}}
%\def\dH{\color{green} H \color{black}}

\def\cP{{\mathcal P}}

\def\Lint{{\mathcal P}}

\def\Lintn{{\mathcal P} \cap \gtupn}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
\Header{ \vb 2004 NEW}

\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\Header{\vb Systems problems $\rightarrow $ Matrix Ineq}
\begin{slide}

\vspace*{.2cm}
\begin{center}
\unitlength=0.1mm
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\put(100,100){\framebox(160,85){}}
\put( 60,160){\vector( 1, 0){ 40}}
\put( 60,120){\vector( 1, 0){ 40}}
\put(175,145){\makebox(0,0){{  { Given}}}}
\put(260,160){\vector( 1, 0){ 40}}
\put(260,120){\vector( 1, 0){ 40}}
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\put(135,60){\vector(-1,0){75}}
\put(180,60){\makebox(0,0){{  {\Find}}}}
\end{picture}
\end{center}
\vspace{.4cm}

Many such problems Eg. $H^\infty$ control


%%%%%%%%%%%%%%%
\iffalse  OUT MTNS


\begin{tabbing}
Freq. domain \quad \= Nev-Pick ~~~ \= \kill
Freq. domain \> \textcolor{green}{Nev-Pick} \> \quad 1975-85 \\
State space \> \textcolor{green}{tricks} \> Riccatis 1985-95\\
Matrix ineqs \> \textcolor{green}{tricks} \> LMIs \quad  1990 -
\end{tabbing}


Solve numerical LMIs,
{ \small up to  $150 \times 150$ (unstructured) matrices},
via semidef prog., %2nd order cone prog.,
bundle methods, etc.  \ \
$$
%\begin{equation*}
\left.
\begin{array}{lcr}
F_1(\efX) := A \efX + \efX A^T + BB^T & \succeq & 0 \\
F_2(\efX, \efY) & \succeq & 0
\end{array} \right\} \qquad
\text{PosSD}
%\end{equation*}
$$



%%%%%%%%%%%%%%%%%%%%%%%


\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\Header{MATRIX INEQUALITIES }
\begin{slide}

    Polynomial or Rational function of matrices are PosSDef.

%\medskip

IN MTNS
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\fi
%
{\small{
Example:\  Get
Riccati expressions like
$$
 A \unkX + \unkX A^T -  \unkX BB^T \unkX  + CC^T \succ 0
$$
OR linear matrix inequalities Linear Matrix Inequalities like
$$
\left(
\begin{array}{cc}
A \unkX + \unkX A^T + C^T C &  \unkX B\\
B ^T\unkX & I
\end{array}\right) \succ 0
$$
which is equivalent to the Riccati inequality.
%
%Here $A, B, C$ are typically given matrices and $\unkX = \unkX^T$.
%is a
%symmetric matrix which must be found by some numerical
%method.
}}
\ \ \ \ \ \ \ \ \ \ \ \ \ \mtns04
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Header{CONVEX MIs vs LMIs ?}
\begin{slide}

\begin{center}
{\Large Convex is good!}
\end{center}


%%%%%%%%%%%%       %%%%%%%%%%%%%%%%%%%% OUT
\iffalse

\color{red}{\bf
How can we determine easily if a matrix expression is convex?
}
\end{center}


{\bf Prevailing Methods $=$ \color{blue}Construction tricks: \color{black}}

\begin{itemize}
\item[a.]  Schur Complements  to get Linear Matrix I's.
\item[b.] Add variables to get LMI's (not in this talk).
\item[c.]  Set problem up so it tends to produce LMIs.
\end{itemize}

IN
%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\fi


%%%%%%%%%%%
\iffalse
\begin{center}
\color{blue}{\bf
How can we determine if a matrix inequality is equivalent to an LMI?
Symbolically compute that LMI?

\bigskip

}
\end{center}
\fi
%%%%%%%%%%%%%%%%%%%

Given a  function $r$ and the associated {\color{blue}Matrix Inequality}
$$\gamma I  - \dr{r}(\uX) \succ 0. $$
is there an {\color{blue}Linear Matrix Inequality}
$$
 \gamma \Lambda_0   \ - \  \cL(\uX)  \succ 0.
$$
whose
 solutions $\uX$
 are the same as (a component of) the solutions to
$\gamma I  - \dr{r}(\uX) \succ 0$?
\centerline{{\color{blue}
 Some type of convexity for $r$ is necessary.
}}
\ \ \ \ \ \ \mtns04
\end{slide}
%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}

\Header{\vb DIMENSIONLESS FORMULAS}
{\ }
\vspace{-.7cm}


\begin{enumerate}

%%%%%%%%%%%%%%%%%%%%%%
\iffalse OUT

\item
\color{red}OUTER SCALABLE\color{black}\ - NOT THIS
TALK -  Have \color{green}``one" type of  system \color{black} and
protocols for joining them together  to form large networks without
destroying stability.  \color{green} ``One" system many
topologies\color{black}.
\newline -- {\footnotesize F. Kelly, G. Vinnicombe, T. Basar,
Low, Paganini, Doyle}

IN
%%%%%%%%%%%%%%%%%%%%%%%
\fi
\item \color{red} DIMENSIONLESS FORMULAS \color{black} -- THIS
TALK\newline \color{green}
Topology is fixed; but many systems \color{black}.  E.g.

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+$}}
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\put(130,90){\circle{20}}
\put(170,81){\framebox(25,20){}}
\put(179,91){\makebox(0,0){ \color{red}$
S_1$\color{black}}}
\put(195,90){\vector(1,0){20}}
\put(205,90){\line(1,0){20}}
\put(226,81){\framebox(25,20){}}
\put(235,90){\makebox(0,0){ $
S_2$}}
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\put(266,90){\line(1,0){20}}

\put(115,70){\makebox(0,0){ $
-$}}
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\put(130,50){\line(1,0){65}}
\put(195,41){\framebox(25,20){}}
\put(207,50){\makebox(0,0){\color{blue} $
S_3$\color{black}}}
\put(220,50){\line(1,0){55}}
\put(275,50){\line(0,1){40}}
\end{picture}
\vspace{-5mm}

WANT FORMULAS: which hold regardless of the  dimension of
system
\color{red}$S_1 \color{black}, S_2,
\color{blue}S_3$\color{black}.
 E.g. almost all control text problems.

-- Get noncommutative formulas \ \ \ \ \ \ \mtns04
\end{enumerate}
\end{slide}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\Header{DIMENSIONLESS INEQUALITIES}
\begin{slide}

\centerline{ Does Scale Gracefully with Dimension} \vspace{-0.4cm}

\begin{equation}
\label{ineq:lmi1} \left(
\begin{array}{cc}
A \unkX + \unkX A^T + C^TC &  \unkX B\\
B^T\unkX& I\end{array}\right) \succeq 0
\end{equation}
Looks the same regardless of system size.

%\vspace{0.5cm}

\begin{center}
\vspace{-.3cm}
--------------------------------------------------------------
\vspace{-.3cm}

{ Dimension Dependent}
\end{center}

If dimensions of the matrices $A, B, C, \unkX$ are specified
write (\ref{ineq:lmi1})  with matrices $L_0, \dots, L_m$ as,
% LMI FORMULA 5 lines
\vspace{-.5cm}
\begin{equation}
\label{ineq:Ugh} \sum\limits^m_{j=0}\ L_j \unkxj \succeq 0
\end{equation}
 unknown numbers $\unkx_j$,
 ( {\tiny{ from
 $\unkX =\left(
\begin{array}{cc}
\unkxone &  \unkxtwo \\
\unkxtwo  & \color{red} x_3 \color{black}
\end{array}
\right)
$}} )
\end{slide}
%%%%%%%%


%%%%%%%%%%%%


\begin{slide}
\Header{ THE RANGE OF QUESTIONS  }


\centerline{ \color{blue} { Characterize and analyze these
situations:  } \color{black} }
$$
\begin{array}{ccc}
             & LMI  & CONVEX \\
   DIMENSIONLESS (NC) &
\color{red} ? \color{black} &
\color{red} ? \color{black} \\
   DIMENSION \ DEP. (COM)&
\color{red} ? \color{black} & usual \ \ convexity
\end{array}
\ \  \  $$
\bigskip
\bigskip

\color{blue}
\begin{center}{
{\color{red} Q?}
Can we treat many more problems with convex
techniques than LMI techniques?
}
\end{center}

\color{black}

\bigskip
\centerline{ \color{red} {Ultimately we want Symbolic Algorithms}
\color{black} }
\ \ \ \ \ \ \mtns04
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%


%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%

\Header{\vb NOTATION}
\begin{slide}

C{\color{red}O}L{\color{red}O}RS:

Black $=$ knowns  \ \  $A, B$

Colors like {\color{red}red}
{\color{green} green} are variables \ \ \  $\unkx$ \ $\dh$

Caps are matrices   \ \ \ $\unkX$ \ $\dH$

Substitute the matrix $\unkX$ for variable $\unkx$ is \ \
$\unkx \to \unkX$

g-tuples of variables \ \
$\ux = \{ \unkx_1, \cdots, \unkx_g \}$


g-tuples of symmetric matrices \ \
$\uX = \{ \unkX_1, \cdots, \unkX_g \}$; \ \ $\uX \in \gtupn$.

Polynomial
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mtns04
$$
p( \unkx_1, \unkx_2 ): =\unkx_1^4 + \unkx_2^4 + 7 \unkx_1
$$
\end{slide}
%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%

\Header{\vb NONCOMMUTATIVE CONVEXITY}
\begin{slide}
{\large\bf Geometrically matrix convex} \quad
   $0 \leq \alpha \leq 1$
\begin{equation*}
\begin{split}
\rex{r}(\alpha \ \uX +
(1-\alpha) \
\uY) \ \
\preceq \ \
\alpha \ \rex{r} (\uX) +
(1-\alpha) \ \rex{r}
(\uY)
\end{split}
\end{equation*}
{\large\bf Matrix Convexity:}
Plugging any matrices into Hessian $r''(\ux)[\updh]$ of
$\rex{r}$ wrt $\ux$ gives a PosSD matrix.

\smallskip
\centerline{\shadowbox{\color{blue}
\parbox{8.5cm}{{\bf Geometric and matrix convexity are the same,
provided
$\rex{r}(\ux)$ is smooth and there are no constraints on $\ux$.}
   \quad  (cf. Merino+H, CDC1997) }  }}


{\scriptsize
{\bf Notation:}~~~
$r''(\unkxone,\dots,\unkxm )[\updh_1,\dots,\updh_m]$
}
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\Header{ ONE $R^{2 \times 2}$ TEST OF CONVEXITY}
\begin{slide}

%{\small{
{\ }
\vspace{-.6cm}
$$p(\unkx): =\unkx^4 + (\unkx^4)^T$$
  \begin{equation*}
\demi
p(\efX) +
\demi
p({\efY)} -
p(\demi \efX +
\demi
{ \efY} )
\succeq 0 ? \quad \ {\mbox{Try matrices}}
\end{equation*}

  $
   \efX=
  \left(\begin{array}{cc}
    1 & 1 \\
    0 & 1 \
  \end{array}
  \right)
  ,
   \efY=
  \left(\begin{array}{cc}
    1 & 3 \\
    2 & -1 \
  \end{array}
  \right),
$
$
\demi \efX + \demi \efY=
  \left(\begin{array}{cc}
    1 & 2 \\
    1 & 0 \
  \end{array}
  \right)
$%}}
{\tiny{
$$
\demi p( \efX  ) =
\demi \left[
  \left(\begin{array}{cc}
     1 & 1 \\
    0 & 1 \
  \end{array}
  \right)^4    + \left(\begin{array}{cc}
    1 & 0 \\
    1 & 1 \
  \end{array}
  \right)^4
 \right]
  =
   \left(\begin{array}{cc}
    1 & 2 \\
    2 & 1 \
  \end{array}
  \right)
$$
$
\demi p( \efY ) =
  \left(\begin{array}{cc}
    49 & 0 \\
     0 & 49 \
  \end{array}
  \right)
$
  and
$
p(\demi \efX + \demi \efY ) =
  \left(\begin{array}{cc}
    22 & 15 \\
    15 & 12 \
  \end{array}
  \right)
$.
  }}
Then
$$
  \left(\begin{array}{cc}
    1 & 2 \\
    2 & 1 \
  \end{array}
  \right)
   +
   \left(\begin{array}{cc}
    49 & 0 \\
     0 & 49 \
  \end{array}
  \right)
  -
  \left(\begin{array}{cc}
    22 & 15 \\
    15 & 12 \
  \end{array}
  \right)
     \succeq 0 \quad\quad {\mbox{i.e., is PSD}}
    $$

%   {\tiny{
%That  is, it is  PsD (Pos Semi Def).}}%}}
%\begin{equation*}
%\demi p(\uX) + \demi p({\efuY)} -
%p(\demi \uX + \demi { \efuY} ) \succeq 0
%\end{equation*}

\end{slide}

%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
\Header{ \vb NC CONVEX POLYNOMIALS}


\noindent
{\color{red}THEOREM:}
\ \ (Ando $g=1$, McCullough-H) \\

 Convex NC polynomials have degree 2 or less.

\bigskip

\bigskip

\bigskip

\centerline{{\Large {\color{red} What Are Convex NC Rational Functions?}}}


\centerline{This Talk}
\mtns04
\end{slide}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
\Header{ \vb NC RATIONALS and DOMAINS}


EXAMPLE: \ \ \
$
F(\unkx,\unky)= ({\unkx}-{\unky}^{-1})^{-1}, \ \ \
\unkx=\unkx^T, \unky=\unky^T
$

is a sym NC rational function in symmetric vars. \\
Domain \ is \ \
$ \{ \unkX, \unkY : \
      {\unkX}-{\unkY}^{-1} \ and \ \unkY
\  are \ invertible \} $

Given a rational expression $\rex{r}(\ux)$ \ \ \
DEFINE: its {\color{blue} formal domain}
$$
\Dfor{r}:= \{ \uX : \rex{r}( \uX ) \ is \ not \ infinite \}
$$
$\frak{r} $ {\color{blue} rational function}
equiv. classes of rational express $r$
$$
\cF_{\frak{r}}:= \cup_{ \{ r \ represents \ \frak{r} \} } \; \Dfor{r}
$$
${\mathcal F}^0_{\frak{r}}$ is the component of  $ \cF_{\frak{r}}$
 the domain  of $\frak{r}$ containing 0.
\ \ \ \ \ \ \mtns04
\end{slide}


%%%%%%%
%%%%%%%%%%%%%%%%%%%%



%%%%%%%%%%%%%%%%%%%%%
\Header{\vb  Convexity Algorithm }

\begin{slide}

\color{blue}
The theory of Camino - H-  Skelton gives an algorithm
\color{black}

\begin{dingautolist}{202}
\item What is the region of convexity for:
   $$F(\unkx,\unky)= ({\unkx}-{\unky}^{-1})^{-1}, ~~~~~
\unkx=\unkx^T, \unky=\unky^T$$


\vspace*{-2mm}
\begin{itemize}
\item[\color{red}\ding{43}] {\large  APPLY:}~~ {\bf
NCConvexityRegion}[$F$, \{$\unkx$, $\unky$\}]
\vspace*{-4mm}
 $$\color{blue}\Updownarrow$$
\vspace*{-7mm}
$$\{ 2 (\unkx -\unky^{-1})^{-1}, ~~ 2 \unky^{-1} \}$$

\medskip
\item[\color{red}\ding{43}] {\large CONCLUDE:}~~ $F(\unkX,\unkY)$ is
{\color{red} matrix convex} %on the region
$${\cal G} := \{(\unkX, \unkY): \unkY >0 ~~~\mbox{and}~~~ \unkX -
\unkY^{-1} > 0 \}$$
\end{itemize}
ODDITY: \ \ Region of convexity for r is same as a  \\
\hspace{1.0in}  component of domain of r.  \ \ \ \ \ \ \mtns04
\end{dingautolist}
%%%%%%%%%
\end{slide}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
\Header{ \vb SOME MATRIX CONVEX RATIONAL EXPRESSIONS}

\begin{equation}
\label{def:butter}
\dr{r}(\ux)= \dr{r}_0 + \dr{r}_1(\ux) + \ell(\ux) \ell(\ux)^T
\end{equation}
  $\dr{r}_0$ is a scalar, $\dr{r}_1$ and
$\ell(\ux)$ are affine linear in $x$.
%
%\begin{equation}
%\label{def:butter}
%\dr{r}(\ux)=
%      C(I_d - [ A_1 \unkx_1 + \cdots + A_g \unkx_g ])^{-1} C^T
%\end{equation}
%
\begin{equation}
\label{def:butter}
\dr{r}(\ux)=
      \Lambda(\ux) (I_d - [A_1 \unkx_1 + \cdots + A_g \unkx_g ])^{-1} \Lambda(\ux)^T
\end{equation}
$A_j$ are symm matrices, $\Lambda(\ux)$ is $1 \times d$  affine linear in $x$.
%
%
%where $J$ is a symmetry,   $A_j$ are symmetric matrices,
 %   $\dr{r}_0$ is a scalar,
%$\ell(\ux)$ is affine linear in $x$ and
%$\Lambda(\ux)$ is affine linear in $x$.

That is,  $\ell$, $\dr{r}_{1}$ and $\Lambda$
are NC  linear pencils with  $\dr{r}_{1}$ a symmetric linear pencil.

{\color{red} $r$ is  NC Convex where
$$
(I_d - [A_1 \unkx_1 + \cdots + A_g \unkx_g ]) \ is \ Pos \ Def
$$}
\centerline{C. K. Li - Mathias   and  Vanderbai
\ \ \ \ \ \ \  \ \ \ \ \ \mtns04
}
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{slide}
\Header{\vb REPRESENTATION EXAMPLE}
$$
F(\unkx, \unky)=( \unkx- \unky^{-1})^{-1} \ \ \  \ is \ convex \ near \ x=2, y=2
$$
Represent $
r(\unkx, \unky)=F(\unkx+2,\unky+2) , \ \ is  \ convex \ near \ x=0, y=0 \; ,
$
in the form
\begin{equation*}
\dr{r}(\ux)=
      \Lambda(\ux)
      (I_d - [A_1 \unkx_1 + \cdots + A_g \unkx_g ])^{-1} \Lambda(\ux)^T
\end{equation*}
ANS: up to normalization \ \ \
$r(\unkx, \unky)=$
\begin{equation*}
\left(\begin{array}{c}1\\0\end{array}\right)^\text{T}
\left[
\left(\begin{array}{cc}
2&1\\1&2
\end{array}\right)
+
\left(\begin{array}{cc}
1&0\\0&0
\end{array}\right)
\unkx+
\left(\begin{array}{cc}
0&0\\0&1
\end{array}\right)
\unky
\right]^{-1}
\left(\begin{array}{c}1\\0\end{array}\right)
\end{equation*}

\end{slide}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\iffalse OUT
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
\Header{ \vb PROOF OF MATRIX CONVEXITY }

\begin{equation}
\label{def:butter}
\dr{r}(\ux)=
      C(I_d - [ A_1 \unkx_1 + \cdots + A_g \unkx_g ])^{-1} C^T
\end{equation}
$A_j$ are symmetric matrices. \ \  Take $g=1$

$\dr{r}''(\unkx)[\dh]=$
$$
\big[
C (I -  A  \unkx )^{-1}
A    \dh \big]\ \
(I -  A   \unkx)^{-1}
 \ \ \big[ A   \dh
(I -  A    \unkx )^{-1} C^T \big]
$$
Plug in matrices \ \ \
$\dr{r}''(\unkX)[\dH]=$
{\small{
$$
\big[
C \otimes I (I -  A \otimes \unkX )^{-1}
A  \otimes  \dH \big]\ \
(I -  A  \otimes \unkX)^{-1}
 \ \ \big[ A  \otimes \dH
(I -  A  \otimes  \unkX )^{-1} C^T \otimes I \big]
$$ }}
is pos semi def where  $(I_d -  A \otimes  \unkX )$ is pos def.\\
 \centerline{\Large \color{red}{ So $\dr{r}$ is matrix convex there. }}
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\fi
%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
\Header{ \vb REALIZATIONS}

We say $\rf{r}$ has a {\color{blue}  butterfly realization} provided
there is a rational expression for $\rf{r}$ of the form
$$
\dr{r}(\ux) = \dr{r}_0 + \dr{r}_1(\ux) + \ell(\ux) \ell(\ux)^T
\quad \quad \quad .
$$
$$ {\color{blue}(BR)} \quad
\quad
\quad   \quad
\quad
\quad  \quad
 +
      \Lambda(\ux) (J -  [A_1 \unkx_1 + \cdots
      + A_g \unkx_g ])^{-1} \Lambda(\ux)^T
$$
 $J^2=I$ , \ \    $A_j$ are symmetric matrices,
  \ \  $\dr{r}_0$ is a scalar.

$\ell(\ux)$  and $\Lambda(\ux)$ and $\dr{r}_{1}$
are affine linear in $x$,
i.e.
NC  linear pencils with  $\dr{r}_{1}$ a symmetric linear pencil.

\bigskip

\centerline{ {\color{blue}
If $J=I$, then $\dr{r}$ is matrix convex where} }

\centerline{
{\small{$
I -  [A_1 \unkX_1 + \cdots      + A_g \unkX_g ]$
{\color{blue} \ is \ Pos \ Semi \ Def }
}}
}

\centerline{
{\small{ {\color{red} Main Thm says this
{\color{blue} (BR)} is  all Convex funcs.}
}}}
\ \ \ \ \ \ \ \ \ \mtns04
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
\Header{ \vb ALL CONVEX NC RATIONALS}


\noindent
\color{red}THEOREM (McCullough Vinnikov +H) \color{black} \
Suppose  $\rf{r}$ is a symmetric noncommutative rational function
in symmetric variables $\ux$     which is
{ \color{blue}{matrix convex near 0} }
\ \  \ {\color{red}{THEN}}
\smallskip

\noindent
{\color{red}{Realization:} } $\rf{r}$ has a minimal unpinned butterfly
 realization $\dr{r}$ with $J$ equal to the identity.
Specifically
{\small{
$$
\dr{r}(\ux)= \dr{r}_0  + \dr{r}_1(\ux)
+ \ell(\ux) \ell(\ux)^T +
     \Lambda(\ux) ( I_d - [A_1 \unkx_1 + \cdots + A_g \unkx_g ])^{-1} \Lambda(\ux)^T,
$$ }}
where $\dr{r}_1$ and $\ell$ are linear in $x$
and $\Lambda(\ux)$ is  linear in $x$.
 %, that is, $\Lambda_0=0$.

\smallskip

%\noindent
% {\color{red}{?? Pure Butterfly Realization}}
%If $\rf{r}$ has a minimal unpinned butterfly realization with $J=I$,
%then $\rf{r}$ has a minimal unpinned pure butterfly realization
%with $J=I$.

\smallskip

\noindent
{\color{red}Singularities:}
The
%minimal unpinned butterfly
realization $r$
%in
%is either pure ($\Lambda_0=0)$ or
%satisfies  $\Lambda(\ux):= \Lambda_0$,
has domain
whose component containing 0, is
\ \ \ \ \ \ \ \ \    \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  \mtns04
%$\Dfor{r}^0$
%${\mathcal F}^0_{\rf{r}}$,
 % $\Lint$
%defined by
%$$
%\mathrm{int}(\mathcal{P}) :=\bigcup\limits_{n>0}
%{int}(\mathcal{P}^n).
%$$
%where
$$
     \Lint:= \; \{ \uX \in \gtup :
(I_{d} - [A_1 \unkX_1 + \cdots + A_g \unkX_g ]) \ is \ Pos \ Def \}.
$$
\end{slide}
%%%%%%%%%%%%
%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
\Header{ \vb COROLLARIES}

{\tiny{
Recall Butterfly Realization:
$$
\dr{r}(\ux)= \dr{r}_0  + \dr{r}_1(\ux)
+ \ell(\ux) \ell(\ux)^T +
  \Lambda(\ux) ( I_d - [A_1 \unkx_1 + \cdots
+ A_g \unkx_g ])^{-1} \Lambda(\ux)^T,
$$
%\vspace{-0.2in}
where $\dr{r}_1$ and $\ell$ are linear in $x$
and $\Lambda(\ux)$ is  linear in $x$.
}}

\noindent
{\color{red} $\dr{r}(x )$ is a Poly:} \ \ (Ando $g=1$, McCullough-H) \\
Convex NC polynomials have degree 2 or less.

\noindent
Idea: \ \ \ \ $
\dr{r}(\ux)= \dr{r}_0  + \dr{r}_1(\ux)
+ \ell(\ux) \ell(\ux)^T +
     \Lambda(\ux)  \Lambda(\ux)^T $


\medskip


\noindent
{\color{red}{ Convexity Region:}}
The function
$\rf{r}$ is
{ \color{blue}{matrix convex near 0} } implies $\rf{r}$ is
 matrix convex on all of
(the 0 component of) the  domain of $\rf{r}$:
{\small
$$ \Lint = \; \{ \uX \in \gtup :
(I_{d} - [A_1 \unkX_1 + \cdots + A_g \unkX_g ]) \ is \ Pos \ Def \}.
$$}
\centerline{{\color{red}Convexity Region $r \ =$ A Component of Domain $r$}}
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
\Header{ \vb LMI REP THEOREM}

\noindent
\color{red}THEOREM \color{black}  McCullough,Vinnikov, Bill) \ \
If $\rf{r}$ is an NC sym rational function
in symmetric variables $x$
which is
{\color{blue}{matrix convex near 0}}, \\
{\color{red}{THEN}} \ \ \  \
there is a NC symmetric linear  pencil $\cL(\ux)$ and a PsD matrix $\Lambda$\\
%$$\cL(\ux) =  I- [\cA_1 \unkx_1 + \cdots + \cA_g \unkx_g ],$$
%with each $\cA_j$  symmetric \\
{\color{red}{SUCH THAT}} \ \
(the component of 0 of) the set
$$
 \{ \uX \in \gtup :  \gamma I  - \dr{r} (\uX) \ is \ Pos  \ Def \}
$$
equals
all solutions $\uX$ to the LMI
$$ \gamma \Lambda_0 - \cL(\uX) \ is \ Neg \ Def.$$
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mtns04
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%


\begin{slide}
\Header{ STATUS OF QUESTIONS  }

\centerline{ \color{blue} {  For  Rational Functions } \color{black} }

\vspace{ -0.4in}
$$
\begin{array}{ccc}
             & LMI  & CONVEX \\
   DIMLESS (NC) &
\color{black}  CONVEX $=$ \color{black} &
\color{black} ButterF Rep \color{black} \\
   DIM \ DEP. (COM)&
   n=2, %\footnote{$VV+H$}
  { \color{red} n > 2 ? }  & usual \ convexity
\end{array}
\ \  \  $$


\color{blue}
\begin{center}{
{\color{red} Q?}
Can we treat  more problems with convex
techniques than LMI techniques?
%{\color{black} ANS: NO MORE}
}
\end{center}
\color{black}

\begin{center}{ \color{black}{A. \ \  FOR DIMENSIONLESS SYSTEMS PROBLEMS
CONVEX $\Leftrightarrow$ LMI
}}

{ \color{red}
This is true under the restrictions of our set up.}
\end{center}

\centerline{ \color{black} {We have some Symbolic Algorithms}
\color{black} }
\ \ \ \ \ \ \mtns04
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
\Header{ \vb PROOF}
Note that the butterfly realization expresses
$$
\dr{r} -\gamma I = \ \ \ Schur \ complement \ of \
 \cL_\gamma(\ux):=
$$
$$
   = \begin{pmatrix}
    - 1 & 0 & \ell(\ux)^T  \\
      0 & -(I - [\cA_1 \unkx_1 + \cdots + \cA_g \unkx_g ]
       )  &  \Lambda(\ux)^T \\
    \ell(\ux) &  \Lambda(\ux) & \dr{r}_0 - \gamma  + \dr{r}_1(\ux)
  \end{pmatrix}
$$
for any real number $\gamma$. \ \
GET
%{\small{
$$
\dr{r}(\uX) -\gamma I \prec 0 \ \ \ iff  \ \ \
\cL_\gamma(\uX) \prec 0
$$
PROVIDED \ \
Singularities of  $\dr{r}(\uX)$ contain
$. \ \ \ \ \ \ \ \ \ the \ boundary \ of \
- ( \; I - [\cA_1 \unkX_1 + \cdots + \cA_g \unkX_g ] \; ) \prec 0$.\\
This is what {\color{red}{Singularities Theorem}} says is true.
%}}
\end{slide}
%%%%%%%%%%%%%%%%



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
\Header{ \vb PROOF OF REALIZATION}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\def\nxn{{n \times n}}

\noindent
{ \color{red}{THM} }
$\rf{r}$ has a
minimal {\color{red}Descriptor  Realization} $\dr{r}$ with $J^2=1$
and $A_j \in \bbR^\nxn$ symmetric.
Specifically
\begin{equation} \label{eq:butterI}
\dr{r}(\ux)= \dr{r}_0
+    C ( J - \sum_j^g A_j \unkx_j )^{-1} C^T,
\end{equation}

\centerline{
{ \color{red}
Schutzenberger, Fliess, Becker, Ball -Malikorn
} }

FISHY "It's too good" Ex. It represents polynomials of degree 10
easily.

\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
\Header{ \vb CONVEXITY vs $J=I$}

\noindent {\color{red}{LEMMA}} \
{ \color{blue}{
Matrix convex $+$ minimal implies
$J=I$

\centerline{
{\color{red} iff} }

there is no pinning vector $\eta$, ie.
$\eta \in \cap_j \ Null(A_j)$. }}

PF: $g=1$  \ \ Substitute all small symm
$\unkX \in \bbR^\nxn$ and all symm $\dH \in \bbR^\nxn $
\ \ \ \ \ \ \ \ \ $
x \to X \  \ \  and \ \  \ \dh \to \dH
$

$0 \preceq \dr{r}''(\unkX)[\dH]=$

\centerline{
{{\tiny
$
\big[
C\otimes I_n (J\otimes I_n -  A \otimes \unkX )^{-1}
A  \otimes  \dH \big]$
{ \small{
$(J \otimes I_n -  A  \otimes \unkX)^{-1}
$ }}
$ \big[ A  \otimes \dH
(J\otimes I_n -  A  \otimes  \unkX )^{-1} C^T\otimes I_n \big]
$
}}
}

Can show \
$\big[ A  \otimes \dH
(J -  A  \otimes  \unkX )^{-1} C^T \big]$ \
sweeps all but "a little" of  $\bbR^{nd}$
$\implies$
$
(J -  A  \otimes \unkX)^{-1}$ is Pos SemiDef,
all $\unkX$ small, say $\unkX=0$.  \ So $J \succ 0$.
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
\Header{ \vb UNPINNED and SINGULARITIES }

If  minimal and convex,
then pinning vectors $\eta$ are a one dim space.
Can show $J$ has at most 1 neg eigenvalue.

This allows conversion
to an unpinned butterfly realization with $J=1$
(several pages of algebra).

Minimality can be gotten by "cutting down".


SINGULARITIES:
{\small{
$$
 \dr{r}( \uX )= \dr{r}_0 + \dr{r}_1( \uX ) + \ell( \uX ) \ell( \uX )^T
 + \Lambda(\uX ) (I -  [A_1 \uX_1 + \cdots + A_g \uX_g ])^{-1}
 \Lambda( \uX )^T
 $$
}}
is singular on the set
{\small{
$$ boundary \ of \
 \{ \uX : (I -  [A_1 \unkX_1 + \cdots + A_g \unkX_g ]) \succeq 0 \}
$$
}}
Proof: \  Pain in the neck.
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
\Header{ \vb NC CO-ORDINATE CHANGES}

\begin{center}
  NONCOMMUTATIVE ANALYTIC \\ CO-ORDINATE CHANGES
\end{center}

\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
\Header{ \vb NC CO-ORDINATE CHANGES}

Variables are not symmetric.

An analytic polynomial map in $g$ variables $\ux$
{\small
$$
\ux \to p(\ux) :=
\left(%
\begin{array}{c}
  p_1(\ux) \\
  \vdots \\
  p_k(\ux) \\
\end{array}%
\right)
$$}
Analytic means $p$ contains no $x_j^T$.
%
An NC symmetric hereditary polynomial
$q $  which is matrix positive, defines
%It defines a domain
$$
\cD_q:= \{ \uX \in (\bbR^{n \times n})^g : \ \ I \succeq q(\uX) \}.
$$
{\color{blue}
Is there a bianalytic map $p$ of

\centerline{
$\cD_q$  onto
the unit ball
$ \cD_{ \unkx_1^T \unkx_1 + \cdots + \unkx_g^T \unkx_g}$?
} }


\end{slide}
%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
\Header{ \vb BiAnalytic Equivalence}
%\def\cB{\mathcal B}

{\color{blue}
\centerline{
Is there a bianalytic map $p$ of
}

\centerline{
$
\cD_q:= \{ \uX  : \ \ I \succeq q(\uX) \}
$ \\
onto
the unit ball
$ \cD_{ \unkx_1^T \unkx_1 + \cdots + \unkx_g^T \unkx_g}$?
}

{\small
Commutative $\mathbb C$ case $g=1$: solved by Riemann Map Thm.
}
}

{\color{red}THEOREM} \  N. Slinglend $+$ H \\
{\color{red}
SUPPOSE}  that $q(\ux)$ is a NC hereditary, matrix positive polynomial
such that $q(0)=0$.
%
%Define $r(\ux):=a-p(\ux)$ and
%$d(\ux):=a-c(\ux)$ where
%$$c(\ux):= \overset{g}{\underset{j=1}{\sum}}\unkxj^T\unkxj.$$

{\color{red} THEN}
We can construct  NC bianalytic map between
$\cD_q$ and $\cD_{ \unkx_1^T \unkx_1 + \cdots + \unkx_g^T \unkx_g}$
 using an LMI  or no such maps exists.

{\color{green}
{\small
Surprising since construction is an LMI
plus a rank constraint. Very nonconvex.
We handle the rank constraint.}
}
\end{slide}
%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
\Header{ \vb Ball Map Theorem}
\def\cB{\mathcal B}

What are the NC analytic maps $p$ of
$$the \  unit \ ball \
 \cD_{ \unkx_1^T \unkx_1 + \cdots + \unkx_g^T \unkx_g}$$
into
$$
the \ unit \ ball \
 \cD_{ \unkx_1^T \unkx_1 + \cdots + \unkx_k^T \unkx_k}$$
 which take the boundary into the boundary? \ $ k \geq g$
We call these  {\color{blue} Joe Maps}

{\small
Commutative case De Angelo 1990ish. Simple ans if $k < \frac{g}{2}$
}


{\color{red}THEOREM} \  N. Slinglend $+$ H \\
\centerline{Every  Ball Map is a linear  isometry.}
\end{slide}
%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
\Header{ \vb NC SEMI-ALGEBRAIC  GEOMETRY}

%%
\begin{itemize}
\item    \ CLASSICAL  \color{orange}  COMMUTATIVE \color{black}
Suppose a polynomial $p$ on ${\mathbb R}^n$  is positive on the domain,
${\cal G} \subset {\mathbb R}^n$,
where another polynomial $\delta$ is positive.
Must there be an algebraic relationship between
$p$ and $\delta$?

\color{red}
Ans YES $=$ Positivstellensatz  1920's - 1975's \\
 (active area today- has applications) \color{black}

\vspace{3mm}

\item    \  \color{red} NONCOMMUTATIVE \color{black}

Does this generalize to polynomials in NC variables
$$\ux= \{ \unkx_1, \cdots, \unkx_g \}$$
\color{red}
Ans Often YES.  McCullough, Putinar, H \  \ \ \ \ \mtns04
\color{black}
\end{itemize}
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
\Header{ \vb ALGORITHMS and IMPLEMENTATIONS }
\begin{enumerate}
\item {\color{red}
Convexity Checker} - Camino, Skelton, H  \ \ \ \ Public
\item
{\color{green}
Realization Builder: Convex Rational to LMI}
- Slinglend, Shopple \ \ \ in progress
\item {\color{blue}
Numerical matrix unknowns} - Camino, Skelton, H  \ \ \ in house
\item {\color{red}
LMI Producer} (uses existing methods on special problems) de Oliveira, H
    \ \ \ \ in house (out soon)

\end{enumerate}

\centerline{ {\color{blue} \Large Try NCAlgebra }}
\ \ \ \ \ \ \ \ \ \mtns04
\end{slide}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
\Header{ \vb END 2004}
END 2004
\end{slide}
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%%%%%%%  FAKE END 1
\end{document}
%%%%%%%  FAKE END 1

\Header{FAKE END I}
\begin{slide}

\centerline{ \Large FAKE END I}

\end{slide}



\begin{slide}
\Header{ \vb MOTIVATION}

{\small
One of the main engineering developments in the 1990's was that systems problems
could be converted to matrix inequalities.  Unfortunately, one often
gets a badly behaved list of such inequalities.  Fortunately, about
2 dozen paradigm systems problems convert to Linear Matrix
Inequalities LMIs.  These are definitely nice and there are  good
numerical solvers but  many problems  are relegated to  compromise.
The techniques are homegrown within the controls community
and one can ask: what branch of mathematics does this
correspond to and what can it do for engineering problems?
}
\end{slide}

\begin{slide}
\Header{ \vb MOTIVATION}

{\small
The classical mathematical subject called real semi-algebraic geometry
deals with polynomials on $R^n$ and inequalities among them,
such as where they are greater than 0.  It forms the underpining
for hybrid control and for many applications recently discovered
by Pablo Parrilo and others.  What is needed is a theory of
inequalities on  polynomials in matrices,
that is, noncommutative polynomials.
To what extent does real semi-algebraic geometry extend to
polynomials on matrices? That is the subject of this talk.
}

\end{slide}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\Header{\vb 1.  "POSITIVE" NC POLYS}
\begin{slide}
$$
{\bf \Large p \geq 0 \ \ \ \ {\mbox{Means What ?}}}
$$

$p$  is a symmetric  polynomial in non-commutative
variables $\ux=\{\unkxone,  \unkxtwo\}$ with real coefficients, eg.
\begin{equation*}
p(\ux)  = \unkxone^2 +  (\unkxone^2)^T   + \unkxtwo^T \unkxtwo
\end{equation*}
{\color{red}{Define}} {\bf \large  MATRIX POSITIVE polynomial}
\begin{center}
Plug in $n\times n$ matrices
$\XXj$ for $\unkxj $
in $p$ \ \

always get

%$\Rightarrow$ \ \
$p(\Xone, \Xtwo)$ is a PosSD $n \times n$  matrix.
\end{center}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\iffalse  ??OUT

\vspace*{-.6\baselineskip}{\footnotesize{
\begin{tabbing}
{\color{red}{Riddle:}}~\= \kill \\
{\color{red}{Riddle:}}\> 1. Is it easy  to
check Mat Pos computationally? \\
\> 2.  Is
   $p$ above Matrix Positive?
\end{tabbing}
}}
IN
\fi
%%%%%%%%%%%%%%%%%%%%%%%
\end{slide}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\iffalse OUT??
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
\Header{\vb MATRIX POSITIVE POLYNOMIALS- Examples}
\begin{slide}

Soln to Ex. 2
\vspace{-.4cm}

\begin{eqnarray*}
p(\ux) & =& \unkxone^2 +  (\unkx^2_1)^T   + \unkxtwo^T
\unkxtwo
\quad \mbox{is NOT
Mat Pos} \\
&& \unkxone \rightarrow\tiny{ \left[
\begin{array}{rr}
         0 & 1 \\ -1 & 0
         \end{array}
\right]}
\quad
\unkxtwo \rightarrow\tiny{ \left[
\begin{array}{rr}
         1 & 0 \\ 0 & 1
         \end{array}
\right] }\\
% \rightarrow \left[\begin{array}{rr}
%       1 & 0 \\ 0 & 1
%       \end{array}\right]
p(\uX) &=& \tiny{ \left[\!\!
\begin{array}{rr}
         0 & 1 \\ -1 & 0
         \end{array}\!\!
\right]
   \left[\!\!
\begin{array}{rr}
         0 & 1 \\ -1 & 0
         \end{array}\!\!
\right]
+
   \left[\!\!
\begin{array}{rr}
         0 & -1 \\ 1 & 0
         \end{array}\!\!
\right]
   \left[\!\!
\begin{array}{rr}
    0 & -1 \\ 0 & 1
    \end{array}\!\!
\right]
   + \left[\!\!
\begin{array}{rr}
    1 & 0 \\ 0 & 1
    \end{array}\!\!
\right]
   \left[\!\!
\begin{array}{rr}
    1 & 0 \\ 0 & 1
    \end{array}\!\!
\right] }\\
&=& \tiny{\left[
\begin{array}{rr}
    -1 & 0 \\ 0 & -1
    \end{array}
\right]}
\end{eqnarray*}


Soln to Ex. 1  \quad
{\footnotesize{
{\color{red}{Validating MP by Brute force horrendous.}} \\
%\begin{quote}
Curse of dimensionality:
Eg.
$\Xone, \Xtwo \in R^{5 \times 5}$ with 40 grid points per dim
is $( 40)^{2 \times 25}$ grid pts.
Requires about $10^{60} $ years .
}}

\end{slide}
%%%% IN
\fi
%\newslide
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
\Header{\vb MATRIX POSITIVE POLYNOMIALS- Examples}
\begin{slide}

Example \quad $\ux= \{ \unkxone, \unkxtwo \} $

\iffalse
\begin{equation*}
\begin{alignat*}{4}
p(\ux) &= (3\unkx^2_1 +\unkxtwo)^T && (3\unkx^2_1 + \unkxtwo) && +
(\unkxone^5)^T && (\unkxone^5)  \\
&   && \Downarrow  &&  && \Downarrow \\
p(\uX) &=  && \text{PosSD} && + && \text{PosSD}
\end{alignat*}
\end{equation*}
\end{footnotesize}

------------------

\fi


\begin{footnotesize}
\begin{equation*}
\begin{array}{ccccc}
p(\ux) &= (3\unkx^2_1 +\unkxtwo)^T & (3\unkx^2_1 + \unkxtwo) & +
(\unkxone^5)^T & (\unkxone^5)  \\
&   & \Downarrow  &  & \Downarrow \\
p(\uX) &=  & \text{PosSD} & + & \text{PosSD}
\end{array}
\end{equation*}
\end{footnotesize}



This is a sum of squares (SoS).
More generally
$$
p(\ux)=
\sum_{j=1}^c \cL_j(\ux)^T  \cL_j(\ux)
$$
is clearly always Matrix Positive.

\end{slide}




\Header{\vb TESTING POLYS FOR POSITIVITY}
\begin{slide}
%\centerline {\bf TESTING POLYNOMIALS FOR POSITIVITY}
%\vspace{3mm}

Write poly $p$ as Sum of Squares (SoS)

\centerline{ $p=$ Sum of Squares (SoS) \ $\Rightarrow p$ \ is pos}
\vspace{3mm}

{\bf  QUESTION}: \qquad  SoS  $\overset{{\huge{?}}}{\Leftarrow} p$ is pos?
\vspace{3mm}

{\color{yellow}{ COMMUTATIVE}}   Poly on $R^n$:  Hilbert 17th problem

\begin{tabbing}
1234\= n \= 1,23334 \= and 1pos2 \= 1212121212 \= \kill
\> $n$ \> $=1$ \> and  pos \> $\Longleftrightarrow$ SoS \> YES \\
\> $n$ \> $\ge 2$ \> and pos \>$\Rightarrow$ SoS  \> NOT ALWAYS
\end{tabbing}

\hspace{2mm} Algorithms (commutative) for SoS: - effective
\begin{quote}
\begin{itemize}
\item Choi -Lam- Reznick (95), Powers - Wormann late 1990's
\item Parrilo's  2001 Thesis $+$ applications;
%\item  MTNS Session
\end{itemize}
\end{quote}

\end{slide}


\Header{\vb NON-COM POLYS ARE BETTER BEHAVED THAN COMMUTATIVE POLYS}
\begin{slide}

\parbox[b]{2cm}{\bf THM:}
\raisebox{-4mm}{\shadowbox{\color{red}\parbox[t]{9.5cm}{\bf \Large
Matrix positive non-commutative polynomials are sums of squares }}}

\bigskip

%Maybe non-commutative inequalities are algebraically better
%behaved than commutative.

False in the commutative case.
Can compute SoS, thus CAN TEST Matrix Positivity.
\smallskip

Proof: Operator theory techniques
\begin{itemize}
\item  See 2001 H preprint.
\item    S. McCullough  2000 preprint - when all variables $\unkxj$ are
``unitary".
\end{itemize}

%\vspace*{10mm} See 2001 H preprint.

\end{slide}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%








%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\Header{\vb p is Positive on a (SID) -Definition }
\begin{slide}


%%%%%%%%%%%%%%%%%%%%%%%%%%

A Symbolic Inequality Domain (SID) is
$$\sidG:=\{\uX : \delta_1(\uX) \ge 0, \dots, \delta_c(\uX) \ge 0  \XjH  \}$$
%Eg. $ \ux=\{  A, C, \unkX  \}$
${\cal L}(H)$ is all bounded operators on Hilbert space $H$.   \\ For example,
$$\sidG :=\{ \uX=(A, C,  \unkX): \  A \unkX + \unkX A^T - C \ge 0, \unkX \ge 0\}$$
%
$p(\ux)$ is {\color{red}{ OPERATOR POSITIVE  on SID $\sidG$}}
means: For  any operator tuple $ \uX$,
satisfying
$$\delta_1(\uX) \ge 0, \dots, \delta_c ( \uX) \ge 0, $$
we have $p(\uX)$ is a Positive Semidefinite  operator.



$$\sidG:=\{\uX : \delta(\uX)  \ge 0  \XjH  \}$$


Example \quad   $p$ is a weighted sum of squares:
$$
p(\ux)=
\sum_{j=1}^c \cL_j(\ux)^T \delta(\ux) \cL_j(\ux)
$$
clearly is OPERATOR POS  on the SID $\sidG $.



\end{slide}

%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
\Header{\vb NC POSITIVSTELLENSATZ}

\parbox[b]{7.2cm}{ \bf THM   (McCullough +H):}
\raisebox{-4mm}{\shadowbox{  \parbox[t]{9.4cm}
{\bf
   If $p$ is ``operator strict pos" on the bded (by $C$) \ SID

\centerline{$\mathcal{G}:=\{\uX: \delta(\uX) \succeq 0, \XjH  \}$  }
then $\exists$ NC polys
$ {\color{blue} \rho_k },  {\color{blue} \mathcal{L}_j },
 {\color{blue} \mathcal{W}_j }$ such that:

\centerline{$p(\ux):= \sum\limits_{k=1}^{finite}
{\color{blue} \rho_k }(\ux)^T
{\color{blue} \rho_k}(\ux)$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }

\centerline{\qquad \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
  $  + \sum\limits^{finite}_{j=1} {\color{blue} \mathcal{L}_j }
   (\ux)^T \delta(\ux)
  {\color{blue} \mathcal{L}_j }(\ux)$  }

\centerline{\qquad  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  $ +
\sum\limits^{finite}_{j=1}
{\color{blue} \mathcal{W}_{kj}(\ux)^T }
( C^2 - \unkx_j^T \unkx_j  ){ \color{blue} \mathcal{W}_{kj}(\ux)}$  }

\centerline{If the set  $\mathcal{G}$ is convex, then
can take $H$ finite dim.}
}} }

\end{slide}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
\Header{\vb NC POSITIVSTELLENSATZ}

PROOF. Follows outline of
M. Putinar + F. Vasilescu. \ IDEA:

RepSet$:=$ All \ Polys \ which \ have \ PosSS \ representation


\color{green}
SUPPOSE $p$ poly not in RepSet:
\color{black}
\ HahnBan, etc  gives $\ell$
$$ \ell(p) \leq 0  \ \ \ \ and \ \  \ \ \ell( RepSet ) \geq 0. $$
Build Hilbert $H, \ ( \cdot, \cdot)$ \\
\centerline{ $H:= \{f \ a \ polynomial \}$ with $(f,g): = \ell(f g^T)$}
$$[\unkX_j f](x):= Mult_{x_j} f(\ux)= x_j f(x)$$
$$[\unkX_j^T f](x):= Mult_{x_j^T} f(x)= x_j^T f(x)$$.

{\tiny{
$(f,g): = \ell(f g^T)$
$$[\unkX_j f](x):=  x_j f(x)  \ \  and \ \
[\unkX_j^T f](x):= x_j^T f(x).$$}}
{\small
Properties:
\begin{enumerate}
\item $(f,f) = \ell(f f^T) \leq 0$
\item  Mod out null space to get $H$.
\item $ ( [ C^2 -  \unkX_j^T \unkX_j  ] f, f) \leq
\ell( [ C^2 -  \unkX_j^T \unkX_j ] f f^T) \geq 0$, \\
\quad \quad \quad so $C \geq \| \unkX_j \|$
\end{enumerate}
BUT:
$$
(\delta(\uX) g,g)= (g^T \delta(\uX) g 1,1) = \ell( g^T \delta g) \geq 0
\Longrightarrow p(\uX) \geq 0 \ \
$$
$$
(p(\uX)1, 1)= \ell( p )< 0 \ \ \ by \ assumption.
$$
\color{green}
Contradiction!
\color{black} ---   but I cheated a lot.
}
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
\Header{\vb NC POSSS}
\vspace{-2.0cm}

\parbox[b]{5.2cm}{\bf THM   \tiny{(McC + Putinar +H)}:}
\raisebox{-4mm}{\shadowbox{\parbox[t]{9.4cm}{\bf
   If $p$ is ``operator  pos"
   on the  \ SID

\centerline{$\mathcal{G}:=\{\uX: \
 I -   \Xone^T \Xone \geq 0, \cdots,  I-\unkX^T_m \unkX_m \geq 0  \},$}
then $\exists$ NC polys  $ {\color{blue} \rho_k}$ such that

\centerline{$p(\uX):= \sum\limits_{k=1}^{finite} {\color{blue} \rho_k }(\uX)^T
{\color{blue} \rho_k(\uX)}$
}
for
$\uX \ in \ \mathcal{G}$.
}}}
False in commutative case.
\end{slide}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
\Header{\vb NO REAL NC NULLSTELLENSATZ}

COMMUTATIVE REAL  NULLSTELLENSSATZ SAYS

THM: Suppose $p, \delta $ are real valued polynomials on $R^n$.
If $p(x)=0$ on all $x$ in $R^n$ on which $\delta(x)=0$,
then
$$ p^{2m} + SoS \ is \ in \ the \ ideal \ I_\delta \ generated  \  by \ \delta$$


\noindent
\color{red}
NC REAL  NULLSS COUNTEREX  {\tiny(McCul $+$ H)}
\color{black}
$$\delta= (\unkx^T \unkx + \unkx \unkx^T )^{2} \ \ and \ \  p= \unkx +\unkx^T$$
$\delta(\unkX)v=0$ implies $p(\unkX)v=0$, but NO such representation is possible.


\noindent
\color{red}
NC   NULLSS IS TRUE: If $p$ and $\delta$ contain no transposes.
  {\tiny(Pf by G. Bergman)}
\color{black}

\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

     \Header{ \color{red} SUMMARY - NONCOMMUTATIVE \color{black}  }

\begin{slide}

\begin{enumerate}
\item DIMENSIONLESS SYS PROBLEMS ARE NC.

\item \smthm \color{red} 2 \ \ \color{black}
\ EVERY  ``POSITIVE" NC POLYNOMIAL
IS A SUM OF SQUARES.
   \color{green}$\leftarrow$ COMPUTABLE. \color{black} \\
(False for commutative polys)

\item \smthm  \color{red} 3 \ \ \color{black}
THMs 2  LOCALIZES, eg. \\
\quad \quad a.  \ NC STRICT POSITIVSTELLENSATZ \\
\quad \quad b. \ A NC NONSTRICT (VAGUER) POSITIVSS
FOR SPECIAL $\delta$. \
(False for commutative polys)


\item \smthm  \color{red} 5 \ \ \color{black}
NC REAL NULLSTELLENSATZ fails.
Radical ideals - bad characterization. \\
(True for commutative polys)

\end{enumerate}
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%

\begin{slide}
\Header{ \vb NEW TOPIC}

\centerline{ \color{red} {\Huge REGIONS OF    } \color{black} }




\vspace{3mm}

\centerline{ \color{red} {\Huge CONVEXITY} \color{black} }

\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
\Header{\vb CONVEX \ SIDS} {\ } \vspace{-.8cm}

%\vspace{-0.5cm}
\color{red} CONJECTURE:
\color{black}
    Suppose $p $ is a symmetric NC  polynomial with SID
$$\mathcal{G}:= \{\uX: p(\uX) \ge 0  \XjH \} $$
which   is convex and bounded,
then
$\mathcal{G}$ has a  defining polynomial $\tilde{p}$ of degree $\leq 2$.
That is,
$$\mathcal{G}:= \{\uX: \tilde{p}(\uX) \ge 0 \XjH \}. $$

\bigskip
\centerline{{\Large{Roughly: Are Convex SIDs all  trivial?}}}
\bigskip
Easier conjecture
\centerline{{\Large{Convex NC symm polynomials are trivial}}}

%has an LMI representation?  That is, there exist $L_0, L_1,
%\cdots, L_m$,
%$$
%\mathcal{G} = \{\uX: L_0 + L_1\unkXone + \dots + L_{m}
 % \unkXm \ge 0\}? $$
\ \ \ \ \ \ \ \ \ \mtns04
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%

\Header{\vb NONCOMMUTATIVE CONVEXITY}
\begin{slide}



\color{red} Question:  \color{black}
  Consider the noncommutative polynomial
  $$p(\ux): =\unkx^4 + (\unkx^4)^T.$$
   Is it matrix convex?


Function $p$ of noncommutative variables
${\ux} := ( \unkxone, \unkxtwo )$ is MATRIX CONVEX  (geometric def.) $0 \le
\alpha \le 1$
\begin{equation*}
\begin{split}
p(\alpha \uX +
(1-\alpha)
{ \uY} )
\preceq
\alpha p(\uX) +
(1-\alpha) p({\uY)}
\end{split}
\end{equation*}

{\quad} Test
convexity on
$\unkX
\in R^{1 \times 1}$. \color{blue}{YES$_1$:}

\unitlength=0.2mm
\begin{picture}(60,40)(10,20)
\put(300,50){\line(1,0){60}}
\put(330,40){\line(0,1){60}}
\qbezier(330,50)(295,50)(300,100)
\qbezier(330,50)(355,50)(360,100)
\put(370,40){\makebox(0,0){ $\unkX$}}
\end{picture}
\color{black}

{\quad} Test
convexity on two matrices
$\unkX, \unkY
\in R^{2 \times 2}$:
\ \ \ \ \ \ \ \ \ \mtns04
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\Header{ ONE $R^{2 \times 2}$ TEST OF CONVEXITY}
\begin{slide}

%{\small{
{\ }
\vspace{-.6cm}
$$p(\unkx): =\unkx^4 + (\unkx^4)^T$$
  \begin{equation*}
\demi
p(\efX) +
\demi
p({\efY)} -
p(\demi \efX +
\demi
{ \efY} )
\succeq 0 ? \quad \ {\mbox{Try matrices}}
\end{equation*}

  $
   \efX=
  \left(\begin{array}{cc}
    1 & 1 \\
    0 & 1 \
  \end{array}
  \right)
  ,
   \efY=
  \left(\begin{array}{cc}
    1 & 3 \\
    2 & -1 \
  \end{array}
  \right),
$
$
\demi \efX + \demi \efY=
  \left(\begin{array}{cc}
    1 & 2 \\
    1 & 0 \
  \end{array}
  \right)
$%}}
{\tiny{
$$
\demi p( \efX  ) =
  \left(\begin{array}{cc}
     1 & 1 \\
    0 & 1 \
  \end{array}
  \right)^4    + \left(\begin{array}{cc}
    1 & 0 \\
    1 & 1 \
  \end{array}
  \right)^4=
   \left(\begin{array}{cc}
    1 & 2 \\
    2 & 1 \
  \end{array}
  \right)
$$
$
\demi p( \efY ) =
  \left(\begin{array}{cc}
    49 & 0 \\
     0 & 49 \
  \end{array}
  \right)
$
  and
$
p(\demi \efX + \demi \efY ) =
  \left(\begin{array}{cc}
    22 & 15 \\
    15 & 12 \
  \end{array}
  \right)
$.
  }}
Then
$$
  \left(\begin{array}{cc}
    1 & 2 \\
    2 & 1 \
  \end{array}
  \right)
   +
   \left(\begin{array}{cc}
    49 & 0 \\
     0 & 49 \
  \end{array}
  \right)
  -
  \left(\begin{array}{cc}
    22 & 15 \\
    15 & 12 \
  \end{array}
  \right)
     \succeq 0 \quad\quad {\mbox{i.e., is PSD}}
    $$

%   {\tiny{
%That  is, it is  PsD (Pos Semi Def).}}%}}
%\begin{equation*}
%\demi p(\uX) + \demi p({\efuY)} -
%p(\demi \uX + \demi { \efuY} ) \succeq 0
%\end{equation*}

\end{slide}

%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5

\Header{Why? Pos of Second NC Deriv. }
\begin{slide}

{\bf Non-commutative rational function}
$$\Gamma(\unkx,\unky) := \unkx^T a \unkx+d \unky d^T +\unkx
\unky\unkx^T, ~~~\unky=\unky^T, a=a^T$$

{\bf Directional derivative wrt $\unkx$, \ \  $D_\unkx\Gamma(\unkx, \unky)
[\updh]:=$}
%\begin{footnotesize}

\begin{eqnarray*}
= \lim\limits_{t\to 0} \frac{1}{t} \left(\Gamma(\unkx+t\updh, \unky) -
\Gamma(\unkx, \unky)\right) = \left. \frac{d}{dt} \
\Gamma(\unkx+t\updh, \unky ) \right|_{t=0}
\end{eqnarray*}
%\end{footnotesize}

\vspace*{-3mm}
{\bf Examples:}
\begin{dinglist}{51}
\item First derivative of $\Gamma(\unkx,\unky)$ in $\unkx$

\begin{small}
\hfill $D_{\unkx}\Gamma(\unkx, \unky) [\updh] =
\updh^T a \unkx+\unkx^T a \updh + \updh \unky \unkx^T + \unkx \unky
\updh^T$
\end{small}

\item  Second derivative of $\Gamma(\unkx,\unky)$ in $\unkx$

\begin{small}
\hfill ${\cal H}_\unkx\Gamma(\unkx, \unky) [\updh]
= 2 \updh \unky \updh^T + 2 \updh^T a \updh$
\end{small}
\end{dinglist}

\end{slide}


%%%%%%%%%%%%%%%%%%%%%%%%%%%


\Header{Convexity- Two Definitions }
\begin{slide}

{\large\bf Geometrically matrix convex} \quad
   $0 \leq \alpha \leq 1$
\begin{equation*}
\begin{split}
\Gamma(\alpha \uX^{\color{red}1\color{black}} +
(1-\alpha)
\uX^{\color{red}2\color{black}})
\preceq
\alpha\Gamma (\uX^{\color{red}1\color{black}}) +
(1-\alpha) \Gamma
(\uX^{\color{red}2\color{black}})
\end{split}
\end{equation*}
{\large\bf Matrix Convexity:}
Plugging any matrices into {Hessian $\hess{\ux}{\updh}$} of
$\Gamma$ wrt $\ux$ gives a PosSD matrix.

\smallskip
\centerline{\shadowbox{\color{blue}
\parbox{8.5cm}{{\bf Geometric and matrix convexity are the same,  provided
$\Gamma(\ux)$ is smooth and there are no constraints on $\ux$.}
   \quad  (cf. Merino+H, CDC1997) }  }}

%{\large
%Convex on a SID is analogous.  }

{\scriptsize
{\bf Notation:}~~~
$\hess{\unkxone,\dots,\unkxm}{\updh_1,\dots,\updh_m}$

}
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

     \Header{ \color{red} CONVEX SETS (SID) \& FUNCTIONS \color{black}  }

\begin{slide}

\begin{enumerate}


\item  \smthm \color{red} C2 \ {\tiny{ ( ANDO $g=1$, MCCUL $+$ H )}}
 \ \ \color{black} \ \\
 EVERY  NC POLYNOMIAL
 "CONVEX ON AN OPEN SET" HAS DEG $ \leq 2$.
  {  \color{green} Equiv. to a LMI. \color{black} }


\item \smthm \color{red} C1 \ {\tiny{ ( MCCUL $+$ H )}}
 \ \ \color{black} \ \\
  Suppose $p $ is a  polynomial with SID
$$\mathcal{G}:= \{\uX: p(\uX) \ge 0 \XjH \} $$
 in $C^N$.
 Suppose $\mathcal{G}$ is convex ``bded"   and $p$ has form,
%\newline
%\centerline{
\quad $  p(\ux) = \sum\limits^N a_{w}
 {\uX}^{w^T} {\uX}^{w}.$ \ Then  $p$ has degree $\leq 2$.
%}



\item  {  \color{red}  {\tiny{ALGORITHM \ (CHSY) }} \color{black} }
CHECKS  CONVEXITY OF \\ NC RATIONAL FUNCTION.
  \color{green} TRY NCAlgebra \color{black} \\
  {\tiny{Symbolic computer experiments support our conjecture}}


\end{enumerate}
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\Header{Noncommutative LMI's}

\begin{slide}

\noindent
   Example: \ \
%Take $m=2$ and d $=2$
%\begin{equation}
%\label{eq:Ldef1}
% L_1: = \big(
%\begin{array}{cc}
%  2 & 3 \\
%  3 & 0
%\end{array}
%\big)   \  \ \ \ \ \ \ \ \ \
%L_2: =
%\big(
%\begin{array}{cc}
%  3 & 5 \\
%  5 & 0
%\end{array}
%\big)
%\end{equation}
%Then
%$$L_1 \otimes \unkXone=
%L_1: = \big(
%\begin{array}{cc}
%  2 unkX_1 & 3 unkX_1 \\
%  3 unkX_1 & 0
%\end{array}
%\big)   $$
%
%
Take $X_j^T =X_j$. One noncommutative LMI is
$$ L(\ux)=
\big(
\begin{array}{cc}
1 +  2 \unkxone + 3 \unkxtwo  & \ \ 3 \unkxone + 5 \unkxtwo \\
    3 \unkxone + 5 \unkxtwo & 1
\end{array}
\big)\succeq 0.
$$
Clearly, if we were to replace $\ux$ by 2-tuples $\uX =
(\Xone, \Xtwo )$ of  $3 \times 3$ matrices $L(\uX)$ would be
a $6 \times 6$ matrix.

Consider {{Symbolic Inequality Domain, SID, } }
$$\mathcal{G}:= \{\uX: L(\uX) \ge 0 \ \XjH \} $$
{\color{red}
Which convex subsets of tuples from ${\cal L}(H)$ have such a representation?
\ LMI REPRESENTATION
}
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\iffalse  OUT


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{slide}
\Header{\vb $\infty$ EXTRAPOLATE $ \infty$}


%\vspace{-0.5cm}
\color{red} SPECULATION: \color{black}
    Suppose $p $ is an NC symmetric  rational function with SID
$$\mathcal{G}:= \{\uX: p(\uX) \ge 0 \ \XjH \ \  \unkX_j^T = \unkX_j \} $$
which   is convex and bounded,
then does $\mathcal{G}$ have an  NC LMI representation?
I.e, does  there exist $L_0, L_1, \cdots, L_m$,
$$
\mathcal{G} = \{\uX: L_0 + L_1\unkXone + \dots + L_{m}
  \unkXm \ge 0\}? $$


\color{red} ENGINEERING SPECULATION: \color{black}
{\color{blue}{ All Convex}} systems problems which are dimensionless
can all be  converted to {\color{blue}{ LMI}}.


%%%%%%%%%%%%%
\iffalse
   \parbox[b]{5cm}{\bf THMS (McCullough +H):}
\raisebox{-4mm}{\shadowbox{\color{red}\parbox[t]{9.5cm}{\bf
  Yes, with
$L_0, L_1, \dots, L_{m}$ which are ``infinite matrices" }}}

   \parbox[b] {0cm}{}
   %{2cm}{\bf THM:}
   \raisebox{-4mm}{\shadowbox{\color{red}\parbox[t]{10.5cm}{\bf

Yes, Suppose $\mathcal{G}$ is ``bounded"   (in $C^N$) and $p$ has
form,
%\newline
\centerline{$  p_1(\ux) = \sum\limits^N a_{w}
\stackrel{\rightarrow}{X}^{w^T}\stackrel{\rightarrow}{X}^{w}.$
\newline
Then  $p$ has LMI rep. (FinDim)
%degree 2.
}}} }
IN%%%%%%%%
\fi

\end{slide}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{slide}
\Header{\vb COMMUTATIVE  LMI REPS}


%\vspace{-0.5cm}
\color{red} QUESTION (Parrilo- Sturmfels 2001): \color{black}
    Suppose $p $ is a  symmetric  RATIONAL function on $R^{n \times n}$
$$\mathcal{G}:= \{\ux \in R^g: p(\ux) \ge 0 \} $$
which   is convex.
Does $\mathcal{G}$ have an LMI representation?
That is, does there exist $L_0, L_1, \cdots, L_m$,
$$
\mathcal{G} = \{\ux: L_0 + L_1\unkxone + \dots + L_{m}
  \unkx_g \ge 0  \}? $$

ANS: \ (V Vinnikov $+$ H)
\color{red}
 NO there is an
{\bf added } \\ . \quad \quad \quad condition
$\mathcal{G} $  and $p$ must satisfy.
\color{black}
\end{slide}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%





\Header{THE END}
\begin{slide}

\centerline{ \Large THE END}

\end{slide}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%


%\Header{WRAP UP}

\Header{MISC CONVEX SLIDES }

\begin{slide}


\color{red}
\centerline{ \Large MISC CONVEX}


\centerline{ \Large SLIDES}
\color{black}

\end{slide}

%%%%%%%%%%%%%%%%%%%%%%%%%%

\Header{\vb NONCOMMUTATIVE CONVEXITY}
\begin{slide}



\color{red} Question:  \color{black}
  Consider the noncommutative polynomial
  $$p(\ux): =\unkx^4 + (\unkx^4)^T.$$
   Is it matrix convex?


Function $p$ of noncommutative variables
${\ux} := ( \unkxone, \unkxtwo )$ is MATRIX CONVEX  (geometric def.) $0 \le
\alpha \le 1$
\begin{equation*}
\begin{split}
p(\alpha \uX +
(1-\alpha)
{ \uY} )
\preceq
\alpha p(\uX) +
(1-\alpha) p({\uY)}
\end{split}
\end{equation*}

{\quad} Test
convexity on
$\unkX
\in R^{1 \times 1}$. \color{blue}{YES$_1$:}

\unitlength=0.2mm
\begin{picture}(60,40)(10,20)
\put(300,50){\line(1,0){60}}
\put(330,40){\line(0,1){60}}
\qbezier(330,50)(295,50)(300,100)
\qbezier(330,50)(355,50)(360,100)
\put(370,40){\makebox(0,0){ $\unkX$}}
\end{picture}
\color{black}

{\quad} Test
convexity on two matrices
$\unkX, \unkY
\in R^{2 \times 2}$:

\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\Header{ ONE $R^{2 \times 2}$ TEST OF CONVEXITY}
\begin{slide}

%{\small{
{\ }
\vspace{-.6cm}
$$p(\unkx): =\unkx^4 + (\unkx^4)^T$$
  \begin{equation*}
\demi
p(\efX) +
\demi
p({\efY)} -
p(\demi \efX +
\demi
{ \efY} )
\succeq 0 ? \quad \ {\mbox{Try matrices}}
\end{equation*}

  $
   \efX=
  \left(\begin{array}{cc}
    1 & 1 \\
    0 & 1 \
  \end{array}
  \right)
  ,
   \efY=
  \left(\begin{array}{cc}
    1 & 3 \\
    2 & -1 \
  \end{array}
  \right),
$
$
\demi \efX + \demi \efY=
  \left(\begin{array}{cc}
    1 & 2 \\
    1 & 0 \
  \end{array}
  \right)
$%}}
{\tiny{
$$
\demi p( \efX  ) =
  \left(\begin{array}{cc}
     1 & 1 \\
    0 & 1 \
  \end{array}
  \right)^4    + \left(\begin{array}{cc}
    1 & 0 \\
    1 & 1 \
  \end{array}
  \right)^4=
   \left(\begin{array}{cc}
    1 & 2 \\
    2 & 1 \
  \end{array}
  \right)
$$
$
\demi p( \efY ) =
  \left(\begin{array}{cc}
    49 & 0 \\
     0 & 49 \
  \end{array}
  \right)
$
  and
$
p(\demi \efX + \demi \efY ) =
  \left(\begin{array}{cc}
    22 & 15 \\
    15 & 12 \
  \end{array}
  \right)
$.
  }}
Then
$$
  \left(\begin{array}{cc}
    1 & 2 \\
    2 & 1 \
  \end{array}
  \right)
   +
   \left(\begin{array}{cc}
    49 & 0 \\
     0 & 49 \
  \end{array}
  \right)
  -
  \left(\begin{array}{cc}
    22 & 15 \\
    15 & 12 \
  \end{array}
  \right)
     \succeq 0 \quad\quad {\mbox{i.e., is PSD}}
    $$

%   {\tiny{
%That  is, it is  PsD (Pos Semi Def).}}%}}
%\begin{equation*}
%\demi p(\uX) + \demi p({\efuY)} -
%p(\demi \uX + \demi { \efuY} ) \succeq 0
%\end{equation*}

\end{slide}

%%%%%%%5%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\Header{RIDDLE}
\begin{slide}

\begin{enumerate}
\item[\color{red} Q1. \color{black}]
Is
$$p(\unkx): =\unkx^4 + (\unkx^4)^T ?$$
matrix convex?
How would you prove it?

\item[\color{red} Q2. \color{black}] Two vars $\ux= (\unkxone, \unkxtwo )$. Is
$$
p(\ux): = \unkxone^4 + (\unkxone^4)^T
+ 5 \unkxtwo^4 +  5 ( \unkxtwo^4)^T + \unkxone^T \unkxtwo + \unkxtwo^T \unkxone
$$
matrix convex?
How would you prove it?


\item[
\color{red} Q3. \color{black}]  List all matrix convex polynomials that you can
think of, e.g. degree $\le 2$ is easy.
\end{enumerate}

Recall \color{red}MATRIX CONVEX \color{black}
means
\begin{equation*}
\begin{split}
p(\alpha \uX +
(1-\alpha)
{ \uY} )
\preceq
\alpha p (\uX) +
(1-\alpha) p
({\uY)}
\end{split}
\end{equation*}
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}
\Header{\vb 3. $p= X^4 + X^{T^4}$ IS NOT CONVEX }
%\vspace{-.8cm}

\nin
Idea of Proof: \quad \quad  $ p= \unkx \unkx \unkx \unkx + (\unkx
\unkx \unkx \unkx    )^T $

  $D_{\unkx}p(\unkx) [\updh]=$
$$ \updh \unkx\unkx \unkx + \unkx\updh \unkx\unkx
+ \unkx \unkx\updh \unkx  + \unkx\unkx\unkx\updh + ( same )^T
$$

\begin{small}
${\cal H}_\unkx p(\unkx) [\updh] = \ \ \ \
  2 \updh \updh \unkx \unkx +   2 \updh \unkx \updh  \unkx  + etc$
\end{small}

Convex $p$ says its Hessian is  SoS:
$$
{\cal H}_\unkx p(\unkx) [\updh] =
\Sigma_j \
\unkcL_j(\unkx)[\updh]^T  \unkcL_j(\unkx)[\updh].
$$
The highest order terms of $\unkcL_j(\unkx)[\updh]$ are linear in $\updh$.
So
$\updh \unkx \unkx   $ is a term of $\unkcL_j(\unkx)[\updh]$.
Thus the degree 6 term
$\unkx^T \unkx^T  \updh^T \updh \unkx \unkx   $
is a term of the degree 4 polynomial
${\cal H}_\unkx p(\unkx) [\updh] $. \ \ \
{\color{red}{Contradiction. }}\qed
\end{slide}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%
\Header{\vb  Convexity Algorithm }

\begin{slide}

\color{blue}
The theory in the talk gives an algorithm
\color{black}

\begin{dingautolist}{202}
\item What is the domain of convexity for:
   $$F(\unkx,\unky)= ({\unkx}-{\unky}^{-1})^{-1}, ~~~~~
\unkx=\unkx^T, \unky=\unky^T$$


%\vspace*{-4mm}
\begin{itemize}
\item[\color{red}\ding{43}] {\large  APPLY:}~~ {\bf
NCConvexityRegion}[$F$, \{$\unkx$, $\unky$\}]
   $$\color{blue}\Updownarrow$$
\vspace*{-4mm}
$$\{ 2 (\unkx -\unky^{-1})^{-1}, ~~ 2 \unky^{-1} \}$$

\medskip
\item[\color{red}\ding{43}] {\large CONCLUDE:}~~ $F(\unkx,\unky)$ is
{\color{red} matrix convex} on the domain
$${\cal G} := \{(\unkx, \unky): \unky >0 ~~~\mbox{and}~~~ \unkx -
\unky^{-1} > 0 \}$$


\end{itemize}
\end{dingautolist}
%%%%%%%%%
\end{slide}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\Header{ Convexity Algorithm -In Vivo}
\begin{slide}
\( \begin{array}{rl}
{\rm In[1]}:= &  << {\tt NCAlgebra.m;} \\
{\rm In[2]}:= &  << {\tt  NCConvexity.m;} \\[.4cm]
% &  \\
{\rm In[3]} := & {\tt SetNonCommutative[x,y];}\\[.4cm]
%& \\
{\rm In[4]}:= & { \tt F\  =\  inv[x-inv[y]]}\\
%& \\
{\rm Out[5]} := & { \rm inv[x-{inv}[y]]} \\[.4cm]
%&\\
{\rm In[6]}:= & { \tt NCConvexityRegion[F,}\{{\tt x,y}\}{\tt ]} \\ %[.4cm]
%& \\
{\rm  Out[7]}:= & \{\{ {\tt 2 inv[x-inv[y]],\  2 inv[y]} $\}\}$
\end{array}
\)

\begin{center}
\color{blue}
  Download  NCAlgebra :
    \color{black}
  www.math.ucsd $ \sim $ ncalg
   \color{black}
\end{center}

\end{slide}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%


     \Header{ \color{red} SUMMARY  \color{yellow} COMMUTATIVE
\color{black} \color{black} }



\begin{slide}

\begin{enumerate}



\item    \  \color{yellow} COMMUTATIVE \color{black}
  LMI's ( Not Scalable). \\
  \smthm  \color{red} 1 \ \ \color{black}
THERE IS A
{ \color{green}\underline { \color{black}``LINE" TEST}, \color{black} }\
  \ ANY CONVEX
  \color{blue} SET \color{black}
   $\mathcal{C}$ IN
$R^m$ WITH AN LMI REPRESENTATION MUST SATISFY.
IN $R^2$ THIS IS SUFFICIENT.
 Valuable TECHNIQUE \ \
  \color{red} MULTI DIMENSIONAL SYS- WKSHOP and SESS \color{black}

\bigskip
%%%%%%%%%%%%%%%%%%
%\iffalse  OUT

    \color{red} NONCOMMUTATIVE \color{black}
  LMI's ( Scalable). CONJ:The positivity
  \color{blue} set \color{black}
   of an NC polynomial if
   \color{blue} the set \color{black}
    is convex has an LMI representation.

%IN
%\fi
%%%%%%%%%%%%%%%%

\item   \color{yellow} COMMUTATIVE \color{black} SUM OF SQUARES.
HAVE MANY RECENT APPLICATIONS.
  {\color{red} SESSION THIS AFTERNOON  }   \color{black}

\end{enumerate}

\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

     \Header{ \color{red} SUMMARY - NONCOMMUTATIVE \color{black}  }

\begin{slide}

\begin{enumerate}
\item DIMENSIONLESS SYS PROBS ARE NONCOM.


\item \smthm \color{red} 2 \ \ \color{black}
\ EVERY  ``POSITIVE" NC POLYNOMIAL
IS A SUM OF SQUARES.
   \color{green}$\leftarrow$ COMPUTABLE. \color{black}

\item \smthm \color{red} 3 \ \ \color{black}
 EVERY ``CONVEX" NC POLYNOMIAL HAS DEGREE $ \leq 2$.
  {  \color{green} Equiv. to an LMI. \color{black} }

\item \smthm  \color{red} 4 \ \ \color{black}
THMs 2 AND 3 LOCALIZE, eg NC POSITIVSTELLENSATZ




\item  {  \color{red}  {\tiny{ALGORTHIM}} \color{black} }CHECKS  CONVEXITY OF NC RATIONAL FUNCTION.
  \color{green} TRY NCAlgebra \color{black}

\item
 {  \color{red} MAYBE } \color{black}
ALL CONVEX DIMENSIONLESS SYSTEMS PROBLEMS ARE
TREATABLE BY LMIs.
%>AUTOMATICALLY GENERATE THE  LMI?


\end{enumerate}
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%







\Header{THE END}
\begin{slide}

\centerline{ \Large THE END}

\end{slide}




\Header{PLASTIC SLIDES}
\begin{slide}

\centerline{ \Large {PLASTIC SLIDES} }

\end{slide}
%%%%%%%%%%%%%%%%%%%


\Header{DUMP}
\begin{slide}

\centerline{ \Large DUMP}

\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\Header{A Weighted SoS Decomposition}
\begin{slide}

Suppose $Q(\uZ)[\uH] $ is rational in $\uZ$ and quadratic
   in $\uH$.

%\iffalse  OUT
--------------------
\begin{tiny}

Example \quad
$Q[\unkx,\unky][\updH_1,\updH_2]:  =$
\tiny{
$$
\left[\!\!
\begin{array}{c}
\updH_1 (\unkx-\unky^{-1})^{-1} \\ \updH_2  \unky^{-1} (\unkx-\unky^{-1})^{-1}
\end{array}\!\!
\right]^T
\left[\!\!
\begin{array}{cc}
(\unkx-\unky^{-1})^{-1} & (\unkx-\unky^{-1})^{-1} \unky^{-1} \\
   \unky^{-1} (\unkx-\unky^{-1})^{-1} & ~~ \unky^{-1}+  \unky^{-1}
(\unkx-\unky^{-1})^{-1}  \unky^{-1}
\end{array}\!\!
\right]
\left[\!\!
\begin{array}{c}
\updH_1 (\unkx-\unky^{-1})^{-1} \\ \updH_2  \unky^{-1} (\unkx-\unky^{-1})^{-1}
\end{array}\!\!
\right].
$$}
\end{tiny}
%{\footnotesize{
%If $Q(\uZ)[\uH] $ is NOT always PosSemiDef when matrices are plugged in
%for the $Z_j$ and $H_j$, then clearly $Q(\uZ)[\uH] $ is NOT a SoS.
%}}
IN
------------------------
%\fi


We seek a {\color{red}{weighted SoS decomposition}}
$$
Q(\uZ)[\uH]=
\Sigma_j \
%\sum_j
\unkcL_j(\uZ)[\uH]^T \udelta_j(\uZ) \unkcL_j(\uZ)[\uH]
$$
with $\unkcL_j(\uZ)[\uH]$ linear in $\uH$.

%\vspace{-1.0cm}
\parbox[b]{8cm}{ {THM (CHSY)      Under modest hypotheses:    }}
\raisebox{-4mm}{\shadowbox{\color{red}\parbox[t]{9.5cm}{\bf
{{
Any $Q$  has a (computable)
{\color{red}{weighted SoS decomposition}}
with $\unkcL_j(\uZ)[\uH]$ linear in $\uH$ giving SID
$$
\sidG:=\{\uZ : \delta_1(\uZ) \ge 0, \dots, \delta_c(\uZ) \ge 0\}
$$
   which is the ``biggest" SID on which $Q$  is Mat Pos.
}}
}}}
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\Header{NEW TOPIC}
\begin{slide}

\centerline{ \Large CONVEX MATRIX INEQUALITIES}

\end{slide}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Header{CONVEX MATRIX INEQUALITIES}
\begin{slide}

\begin{center}
{\Large Convex is good!}


\color{red}{\bf
How can we determine easily if a matrix expression is convex?
}
\end{center}


{\bf Prevailing methods}

\begin{itemize}
\item[a.]  Use tricks - (Schur Complements) to convert to LMIs.
\item[b.]  Set problem up so it tends to produce LMIs.
\end{itemize}

{\color{red}{ {\bf This Talk} $\rightarrow$}}
\color{black}
Symbolic Computation for Non-commutative Inequalities

{\color{red}{{\bf Broader issue}}}:
What can symbolic non-commutative computation do for systems?
2 Mini-symposia today
\end{slide}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Header{\vb > Is  a  Convexity Checker Possible?}

\begin{slide}

\begin{dingautolist}{202}
\item What is the domain of convexity for:
\begin{small}
$$F( \unkx,\unky)= (\unkx-\unky^{-1})^{-1}, ~~~~~  \unkx=\unkx^T,
\unky=\unky^T$$
\end{small}

\vspace*{-4mm}
\begin{itemize}
\item[\color{red}\ding{43}] {\large APPLY:}~~ {\bf
>NCConvexityRegion}[$F$, \{$\unkx$, $\unky$\}] ?
   $$\color{blue}\Updownarrow$$
\vspace*{-4mm}
$$ \unkx -\unky^{-1},  \quad  \unky $$

\medskip
\item[\color{red}\ding{43}] {\large CONCLUDE:}~~ $F(\unkx,\unky)$ is
{\color{red} matrix convex} on the domain
$${\cal G} := \{(\unkx, \unky): \unky >0 ~~~\mbox{and}~~~ \unkx -
\unky^{-1} > 0 \}$$

We call ${\cal G}$ a {\color{red}{ Symbolic Inequality Domain}}
\color{black}
\end{itemize}
\end{dingautolist}

\end{slide}






     \Header{ > Want Convexity Region Algorithm ?}
\begin{slide}


Region ${{\cal G}}$ in which
$\Gamma $ is convex in $(\unkxone,\unkxtwo)$.
\begin{dingautolist}{202}
\item Symbolic $2^{nd}$ Deriv, the Hessian, is quadratic \newline
   \quad \quad {\small ${\cal Q}(\updHH,\updK) :=
{\cal H}\Gamma(\unkxone, \unkxtwo) [\updHH, \updK]$}
%, $2^{nd}$ deriv. of $\Gamma$.
\item Epress $\Q(\updHH, \updK)$ as $V(\updHH,\updK)^T M_{\Q}
V(\updHH,\updK)$.
%\item Apply the $LDL^T$ decomposition on matrix $M_\Q$, i.e., $M_\Q$ =
%$L D L^T$, to get the diagonal $D$ with non-commutative entries.
\item $M_\Q$ = $L D L^T$, with $D$ diagonal Non-Com entries
$$
\left(
\begin{array}{ccc}
\delta_1(\uAllX) & & \\ & \ddots & \\ & &
\delta_c(\uAllX)
\end{array}
\right)
$$


%\item The Hessian is positive semidefinite on the set ${\cal G}$ of
%$\unkxone$, $\unkxtwo$ which makes the diagonal matrix $D$ positive
%semidefinite. Thus a set $\cG$ where $\Gamma$ is matrix convex is given by
\item  The Hessian is PosSD on the set  ${\cal G}$ of
%$\unkxone$, $\unkxtwo$ given by
\begin{footnotesize}
$$ \cG = \left\{ \uAllX :\quad \delta_j(\uAllX) \ge 0, j =1,\dots,c
\right\}$$
\end{footnotesize}
\end{dingautolist}
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%



%%%%%%%
\Header{The Goals}
\begin{slide}

I. ``Positive" NonCommutative Polynomials

{\color{red}{
\begin{quote}
   What does NonComm positive mean?

How do you compute if a NonComm poly is Positive?
\end{quote}
}}

II. Noncommutative Convexity.

{\color{red}{
\begin{quote}
   What does NonComm convexity mean?

How do you compute where  a NonComm function is Convex ?
\end{quote}
}}

{\color{blue}{What kind of theory is required and exists?}}
\end{slide}



%%%%%%%
%%%%%%%
%%%%%%%

%%%%%%%%%%%%% copy 2 of def of MATRIX POS





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5 FOR BACKUP
\iffalse
%OUT
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
%%%%%%%%%%%%%%%%5555555
\Header{\vb MATRIX POSITIVE POLYNOMIALS- Examples}
\begin{slide}

Example~1
   \footnotesize{
\begin{eqnarray*}
p(\uX) & =& 9\unkXone \unkXtwo + 9 \unkX^T_2\unkX^T_1 +
\unkXone^T \unkXone \quad
\mbox{is not
Mat Pos} \\
&& \unkXone \rightarrow \left[\begin{array}{cc}
         0 & 1 \\ 0 & 0
         \end{array}\right]
\quad
\unkXtwo \rightarrow \left[\left[\begin{array}{cc}
         1 & 0 \\ 0 & 1
         \end{array}\right] \\
p(\uX) &=& 9 \left[\begin{array}
         0 & 1 \\ 0 & 0
         \end{array}\right]
   \left[\begin{array}{cc}
         1 & 0 \\ 0 & 1
         \end{array} \right]
+
9 \left[\begin{array}{cc}
         1 & 0 \\ 0 & 1
         \end{array} \right]
   \left[\begin{array}{cc}
    0 & 0 \\ 1 & 0
    \end{array}\right]
   +\left[\begin{array}{cc}
    1 & 0 \\ 0 & 1
    \end{array}\right]
   \left[\begin{array}{cc}
    1 & 0 \\ 0 & 1
    \end{array}\right] \\
&=& \left[\begin{array}{cc}
    1 & 9 \\ 9 & 1
    \end{array}\right]
\end{eqnarray*}}
\end{footnotesize}
\end{dingautolist}
\end{slide}
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\iffalse
\Header{\vb Convex on a Symbolic Inequality Domain (SID) -Definition DELETE}
\begin{slide}

Convex on a Symbolic Inequality Domain (SIO)

$$\sidG :=\{\uZ) : \delta_1(\uZ) \le 0, \dots,
\delta_c(\uZ)
\le 0\}$$
Eg
$$\sidG :=\{Z unkX + \unkX A + \le 0, \unkX\ge 0\}$$

$\Gamma(\uA, \ux)$ is convex in $\uX$ on
$\sidG $ means: pick any matrices ${{\ESF {\uA_j}}} , \uX$,
%all in $R^{99\times 99}$
satisfying
$$\delta_1(\uA, \uX) \le 0, \dots, \delta_c (\uA, \uX) \le 0 $$
the Hessian of $\Gamma$ at $\uA, \uX$

\centerline{ $\Gamma(\uX) [\updH]$ is PosSD for all matrices $\updH$
   }

Eg


\end{slide}

\fi
%IN
%IN
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\Header{Region of Convexity Algorithm}

\begin{slide}

{{\large The Algorithm }} \quad
{\footnotesize{
$\ux=\{\unkxone, \ldots, \unkxm \}$  $\uH=\{ \updH_1, \ldots, \updH_m \}$
}}

\begin{dingautolist}{202}
\item Let $\Gamma$ be a rational function.

\item Compute the Hessian $\mathcal{H}\Gamma(\ux)[\uH]$ of
$\Gamma$. %The Hessian is quadratic in $\uH$.

\item As the Hessian is quadratic \\ %$\hra$
\hbox{} \dingfill{220}
${\cal H}\Gamma(\ux)[\uH]
=$ $V[\uH]^T M_{{\cal H}\Gamma(\ux)} V[\uH]$.
%and $V[\uH]$ is linear in $\uH$.

\item The non-commutative $LDU$ factorization applied to the
coefficient matrix provides
\quad  \quad $M_{{\cal H}\Gamma(\ux)} = L(\ux)^TD(\ux)L(\ux)$.

\item The resulting diagonal matrix $D(\ux)$ describes the region of
convexity of the function $\Gamma$.

\end{dingautolist}

\end{slide}





\Header{Step by Step thru the Algorithm}
\begin{slide}
Let $B=B^T>0$, $A=A^T$. Define the function $\Gamma(\unkX)$ by
{\footnotesize{
\begin{equation*}
\Gamma(\unkX)= G^T\unkX^TA\unkX
G+\unkX^TB\unkX+G^T\unkX^TC\unkX+\unkX^TC^T\unkX G
\end{equation*}
}}
\vspace{-2.0mm}
The Hessian is given by
\begin{equation*}
\begin{split}
{\mathcal H}\Gamma(\unkX) [\updH] =
2 ( \updH^TB\updH+\updH^TC^T\updH G+ \\
G^T\updH^TA\updH G+G^T\updH^TC\updH ),
\end{split}
\end{equation*}
or equivalently in the form
\begin{equation*}
V[\updH]^T M_{{\mathcal H}\Gamma} V[\updH] = 2  (\updH^T \  G^T\updH^T)
\left( \begin{array}{cc} B & C^T \\ C & A \end{array} \right)
\begin{pmatrix} \updH \\ \updH G \end{pmatrix}
\end{equation*}
\vspace{-11.0mm}
\begin{small}
$
%$Q(\updH ) =
\updH  ( A + G B^T + B G^T + GDG^T ) \updH = $
    commutative case
\end{small}



\newslide
The \LDL\ decomposition with no permutation applied to
$M_{{\mathcal H}\Gamma}$ is
%\begin{footnotesize}
$$
\left(
\begin{array}{cc}
I& 0 \\ C B^{-1} &I
\end{array} \right)
\left(
   \begin{array}{cc}
\textcolor{green}{B}& 0 \\ 0& \textcolor{green}{A - C  B^{-1} C^T}
\end{array}
   \right)
\left(
\begin{array}{cc}
I& B^{-1} C^T \\ 0 & I
\end{array}
\right)
$$
%\end{footnotesize}

The list returned by {\bf NCConvexityRegion} is
   $$ \{ \textcolor{green}{ B, ~~~A - C B^{-1} C^T} \} $$

The necessary and sufficient conditions for the
Hessian to be positive semidefinite are
$$\textcolor{green}{B} >0 ~~~\mbox{and}  ~~~\textcolor{green}{A - C B^{-1}
C^T} \ge 0$$
\end{slide}

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\Header{\vb  Convexity Algorithm Example }

\begin{slide}

%%%%%%%%%%%

\begin{dingautolist}{202}
\item What is the domain of convexity for:
   $$F(\unkX,\unky)= ({\unkX}-{\unky}^{-1})^{-1}, ~~~~~
\unkX=\unkX^T, \unky=\unky^T$$


%\vspace*{-4mm}
\begin{itemize}
\item[\color{red}\ding{43}] {\large  APPLY:}~~ {\bf
NCConvexityRegion}[$F$, \{$\unkX$, $\unky$\}]
   $$\color{blue}\Updownarrow$$
\vspace*{-4mm}
$$\{ 2 (\unkX -\unky^{-1})^{-1}, ~~ 2 \unky^{-1} \}$$

\medskip
\item[\color{red}\ding{43}] {\large CONCLUDE:}~~ $F(\unkX,\unky)$ is
{\color{red} matrix convex} on the domain
$${\cal G} := \{(\unkX, \unky): \unky >0 ~~~\mbox{and}~~~ \unkX -
\unky^{-1} > 0 \}$$


\end{itemize}
\end{dingautolist}
%%%%%%%%%
\end{slide}



\Header{ Convexity Algorithm -In Vivo}
\begin{slide}
\( \begin{array}{rl}
{\rm In[1]}:= &  << {\tt NCAlgebra.m;} \\
{\rm In[2]}:= &  << {\tt  NCConvexity.m;} \\[.4cm]
% &  \\
{\rm In[3]} := & {\tt SetNonCommutative[X,Y];}\\[.4cm]
%& \\
{\rm In[4]}:= & { \tt F\  =\  inv[X-inv[Y]]}\\
%& \\
{\rm Out[5]} := & { \rm inv[X-{inv}[Y]]} \\[.4cm]
%&\\
{\rm In[6]}:= & { \tt NCConvexityRegion[F,}\{{\tt X,Y}\}{\tt ]} \\ %[.4cm]
%& \\
{\rm  Out[7]}:= & \{\{ {\tt 2 inv[X-inv[Y]],\  2inv[Y]} $\}\}$
\end{array}
\)

\end{slide}





\Header{CONCLUSIONS}
\begin{slide}

THM I    Every MatPos Polynomial is a SoS.

THM II  \hangindent=44pt Under modest hypotheses
every
$Q[z][H]$ quadratic  in
$H$ is a weighted SoS whose weights give the
domain of $z$ where
$Q(z) [H]$ is MatPos in $H$.

\medskip

Algo I  There is an algorithm to check MatPos (via
SoS).

Algo II An algorithm based on THM II computes
domain of convexity of a noncom function.  It runs
under NCAlgebra.

\medskip

Algo III You should get NCAlgebra. (Use Google
search engine)

%IV  This conf is fun and thanks to the organizers.

\end{slide}





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\Header{SLIDES NOT ACTUALLY PRESENTED}

\begin{slide}

\begin{center}
{\huge{DOWN WITH INFORMATION OVERLOAD}}
\end{center}

\end{slide}

\Header{AN EXAMPLE: TESTING MAT POS}
\begin{slide}
$$p=1 + XY(X^TX + Y^T Y-1) Y^T X^T$$
\begin{itemize}
\item Substitute 1 dim matrices for $X, Y$
is  $p=1 + ^2 y^2(x^2 +y^2-1)$
Motzkin Poly is pos on $R^2$.

\item Substitute $2\times 2$ matrices for $X, Y$


$X\to \left(
\begin{array}{cc} 0&\alpha\\
0&0\end{array}\right) \quad Y\to
\left(\begin{array}{cc} 0&0\\
\beta&0\end{array}
\right)$

$p=\left[\begin{array}{cc}
1+\alpha^2\beta^2(\beta^2-1) & 0\\
0&1  \end{array}\right]$

Not posSD. when
$0< \beta < 1 \qquad \alpha \gg 1.$

\medskip

\centerline{
Example  due to M. Putinar.}
%}}
\end{itemize}

\end{slide}



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\Header{EXAMPLE: REGION OF CONVEXITY}
\begin{slide}



\begin{dingautolist}{202}
\item Is this matrix function convex (concave) ?
\begin{footnotesize}
$$F(\unkX,\unkY) = -(\unkY+A^T\unkX B)(R+B^T\unkX B)^{-1}(\unkY+B^T\unkX
A)+A^T\unkX A$$ %, ~~\unkX=\unkX^T,~\unkY=\unkY^T$$
\end{footnotesize}

\vspace*{-5mm}
\begin{itemize}
\item[\color{red}\ding{43}] {\large  APPLY:}~~ {\bf
NCConvexityRegion}[$F$, \{$\unkX$, $\unkY$\}]
   $$\color{blue}\Updownarrow$$
\vspace*{-5mm}
$$\{ -2\: (R+B^T\unkX B)^{-1},~ 0,~ 0,~ 0\}$$

%\medskip
\item[\color{red}\ding{43}] {\large CONCLUDE:}~~ $F(\unkX,\unkY)$ is
{\color{red} matrix concave} on %the domain
$${\cal G} := \{ (\unkX, \unkY) : (R+B^T\unkX B)^{-1} > 0\}$$
\end{itemize}

\item Is it the biggest domain of convexity ?

The ``closure'' of $\mathcal{G}$ is the {\Large biggest domain of
convexity}.

\end{dingautolist}

\end{slide}






%%%%%%%%

\Header{\vb  Positivity of Non-commutative Quadratic Functions}
\begin{slide}

%\textcolor{red}{\ding{43}} Determining positivity of the Hessian is
% the key to determining the convexity of a rational function of
% matrices.

\parbox[b]{3cm}{\bf Theorem:} %\hfill
\raisebox{-3mm}{\parbox[t]{3.5cm}{
\shadowbox{\color{red} {\bf Positivity: $\cQ$ {\em versus} $M_\cQ$}}}}
\hfill\hbox{~~~~~~~}

\underline{Assumptions:}
\begin{itemize}
\item $\cQ(\uZ)[\uH]$ is a quadratic in $\uH$;
\item Represent $\cQ(\uZ)$ with coefficient matrix $M_{\cQ(\uZ)}$, and
let $\mathcal{G}$ denote the \textcolor{blue}{\SID}
%\ based on$M_{\cQ(\uZ)}$, namely,
$$\mathcal{G} := \left\{ \uZ: M_{\cQ(\uZ)} \textcolor{red}{\ge} 0\right\}$$
\end{itemize}
Then
$\cQ(\uZ)[\uH]$ is a matrix positive quadratic function for
each $\uZ \in \mathcal{G}$.

\newslide
{\bf Conversely}, assume:
\begin{itemize}
\item[1.] all $\updH $ are restricted to be \textcolor{red}{symmetric};
\item[2.] the coefficients of $V[\uH]$ are \textcolor{red}{linearly
independent};
\item[3.] the domain $\mathcal{G}$ is \textcolor{red}{not thin}
(contains an open set in a certain topology).
\end{itemize}

Then the closure of $\mathcal{G}$ in a certain topology is the biggest
domain on which $\cQ(\uZ)[\uH]$ is  a matrix positive quadratic.
%\end{thm}

\iffalse
{\bf Proof} The sufficient side, the symmetric matrix $M_\cQ$ being
PosSD guarantees that the function $\cQ$ is matrix
positive is obvious.
\textcolor{red}{The necessity side the heavy SoS theory.}
\fi

\end{slide}





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\iffalse

STOP

end

%LAST SLIDE
\Header{\vb Symbolic session at SIAM today \quad Kwatny $+$ H}
\begin{slide}

\quad {\color{blue}{Commutative}:}

{\footnotesize{
Optimizing props of mechanical systems \quad De Jager

Numeric comp for stability of power sys  \quad Fan et. al.

Algebra and convex optimization  \quad Parillio

Integrating Symbolics Graphics Numerics \ \  Rodruiguez - Millan

Tools for nonlinear observer design \quad Kwatny  et. al.
}}

\medskip

\quad {\color{red}{Non-commutative}:}

{\footnotesize{
    Groebner basis applications  \quad  Stankus

    An LMI producing Mathematica notebook \quad \  De Oliveira, Skelton
}}

\end{slide}

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\Header{DUMP}
\begin{slide}
{\Huge{DUMP}}
\end{slide}



%%%%






\Header{The Gram  Representation}
\begin{slide}

{\bf Example:} Quadratic function in $\updH=\updH^T$. \ \ Take $A^T=A$.
\begin{small}
$$\Q(\updH):=
\updH A \updH + G\updH B^T \updH + \updH B \updH G^T + G\updH D\updH G^T$$
\end{small}
This quadratic function can be written in the form
\begin{footnotesize}
\begin{equation}
\Q(\updH ):=
\left( \begin{array}{cc} \updH  & G \updH  \end{array} \right)
\left( \begin{array}{cc} A & B \\ B^T & D  \end{array} \right)
\left( \begin{array}{c} \updH  \\ \updH  G^T \end{array} \right)
\label{e:4.2}
\end{equation}
\end{footnotesize}
A general quadratic $\cQ$ can always be represented as:
$$ \cQ(\updH ) = V[\updH ]^T \textcolor{blue}{M_\cQ}
V[\updH ]$$

$V[\updH ]$  \ding{223} {\bf border ``vector'' of} $\cQ$ ~~~and~~~
$\textcolor{blue}{M_\cQ}$ \ding{224} {\bf coefficient matrix of}  $\cQ$

\end{slide}







\Header{The  Gram Representation}
\begin{slide}

This general notation illustrated by the example (\ref{ex:4.2}) is:
\begin{footnotesize}
\begin{equation*}
V[\updH ]^T=
\left(\begin{array}{cc} \updH  & G^T \updH  \end{array} \right) \qquad
\textrm{and} \qquad M_\cQ = \left(\begin{array}{cc} A & B \\ B^T & D
\end{array} \right)
\end{equation*}
\end{footnotesize}
This contrasts with the {\bf commutative} case where
\begin{footnotesize}
$$Q(\updH ) = \updH  ( A + G B^T + B G^T + GDG^T ) \updH $$
\end{footnotesize}
and the representors are
\begin{footnotesize}
$$M_\cQ= ( A + G^T B + B^T G + G^TDG)  \ \ \mbox{ and } \ \ \ V[\updH
]=\updH .$$
\end{footnotesize}

\vspace*{-4mm}
\textcolor{red}{\ding{43}} The $M_{\cQ}$ representation
%for a quadratic $\cQ$
\textcolor{red}{may not be unique}.
And depends on the particular
quadratic and not only on the number of arguments

%\textcolor{red}{\ding{43}} The size of the $M_\cQ$ representation of a
%quadratic functions $\cQ[\updH , \dots, \updH ]$ depends on the particular
%quadratic and not only on the number of arguments.

\end{slide}
%%%%%%%%%


\Header{Non-Commutative LDU Decomposition}
\begin{slide}

\begin{itemize}
     \item Key tool for determination of matrix positivity of
$M_{\cQ}$, hence for finding the region of convexity $\cal G$.

     \item The LDU factorization applied to an ordinary symmetric
matrix $M$ provides the decomposition $M = LDL^T$, where the matrix
$D$ is diagonal and the matrix $L$ is lower triangular.

     \item Therefore, to check the positivity of the symmetric
matrix $M$ it suffices to check the entries of the diagonal matrix
$D$.

     \item  In the symbolic non-commutative LDU decomposition
one may apply permutation matrices in each step of the process.

\end{itemize}
\end{slide}


     \Header{Convexity Region Algorithm Revisited}
\begin{slide}

\vspace*{1mm}
Provides the region ${\cal G}$ in which $\Gamma(\unkX,\unkY)$ is convex:

\begin{dingautolist}{202}
\item Compute symbolically {\small ${\cal Q}(\updHH,\updK) := {\cal
H}\Gamma(\unkX, \unkY) [\updHH, \updK]$}.

\item Epress $\Q(\updHH, \updK)$ as $V(\updHH,\updK)^T M_{\Q}
V(\updHH,\updK)$.

\item Apply the $LDL^T$ decomposition on matrix $M_\Q$, i.e., $M_\Q$ =
$L D L^T$, to get the diagonal $D$ with non-commutative entries.
\begin{footnotesize}
$$D = \begin{pmatrix} \delta_1(\uZ) & & \\ & \ddots & \\ & &
\delta_c(\uZ)
\end{pmatrix} $$
\end{footnotesize}

\item The Hessian is positive semidefinite on the set ${\cal G}$ of
$\unkX$, $\unkY$ which makes the diagonal matrix $D$ positive
semidefinite. Thus a set $\cG$ where $\Gamma$ is matrix convex is given by
\begin{footnotesize}
$$ \cG = \left\{ \uZ : \delta_j(\uZ) \ge 0, j =1,\dots,c
\right\}$$
\end{footnotesize}

\end{dingautolist}
\end{slide}

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\end{document}
