% Examples of commutative diagrams from Brockett, {\it Finite
% Dimensional Linear Systems}.

\input arrow

\def\Real{\mathord{R}}
\def\Rn{\Real^n}
\def\A{{\bf A}}
\def\B{{\bf B}}
\def\C{{\bf C}}
\def\F{{\bf F}}
\def\G{{\bf G}}
\def\H{{\bf H}}
\def\I{{\bf I}}
\def\K{{\bf K}}
\def\L{{\bf L}}
\def\M{{\bf M}}
\def\N{{\bf N}}
\def\P{{\bf P}}
\def\Q{{\bf Q}}
\def\R{{\bf R}}
\def\S{{\bf S}}
\def\W{{\bf W}}
\def\bp{{\bf\Phi}}


Page 29, Fig.~3
$$\commdiag{\Rn&\mapright^{\bp(t_1,t_0)}&\Rn\cr
&\arrow(3,-2)\lft{\bp(t,t_0)}&\mapdown\rt{\bp(t,t_1)}\cr &&\Rn\cr}$$

Page 48, Fig.~1:
$$\commdiag{\Rn&\mapright^{\bp(t,t_0)}&\Rn\cr
\mapdown\lft{\P(t_0)}&&\mapup\rt{\P^{-1}(t)}\cr
\Rn&\mapright^{e^{\R(t-t_0)}}&\Rn\cr}$$

Page 69, Fig.~1:
$$\commdiag{\Rn&\mapright^{\M_1}&\Real^m\cr
\mapdown\lft\Q&&\mapup\rt\P\cr
\Rn&\mapright^{\M_2}&\Real^m\cr}$$

Page 69, Fig.~2:
$$\commdiag{\Rn&\mapright^\A&\Rn\cr
\mapdown\lft\P&&\mapup\rt{\P'}\cr
\Rn&\mapright^\S&\Rn\cr}$$

Page 71, Fig.~3:
$$\commdiag{\Rn&\mapright^\A&\Rn\cr
\mapdown\lft\P&&\mapup\rt{\P^{-1}}\cr
\Rn&\mapright^\B&\Rn\cr}$$

Page 92, Fig.~1:
$$\harrowlength=40pt \varrowlength=24.7pt \sarrowlength=\harrowlength
\commdiag{C^m&\mapright^{\B(\sigma)}&\Rn&\mapright^{\bp_\A(t_0,\sigma)}&
\Rn&\mapright^{\int_{t_0}^t-d\sigma}&\Rn&\mapright^{\bp_\A(t,t_0)}&
\Rn&\mapright^{\C(t)}&C^q\cr
&\arrow(3,-2)\lft{\P(\sigma)\B(\sigma)}&\mapdown\rt{\P(\sigma)}&&
\mapdown\rt{\P^{-1}(t_0)}&&\mapdown\rt{\P(t_0)}&&\mapdown\lft{\P^{-1}(t)}&
\arrow(3,2)\rt{\C(t)\P^{-1}(t)}\cr
&&\Rn&\mapright_{\bp_{\A_1}(t_0,\sigma)}&\Rn&\mapright_{\int_{t_0}^t-d\sigma}&
\Rn&\mapright_{\bp_{\A_1}(t,t_0)}&\Rn}$$

Page 96, Fig.~2:
$$\harrowlength=32pt \varrowlength=\harrowlength \sarrowlength=\harrowlength
\def\olap#1{\harrowlength=118pt\hbox to0pt{\hss$#1$\hss}}
\commdiag{C^m&&&&&&&&&&C^q\cr \mapup\lft{\G(\sigma)}
&\hmorphposn=2pt\arrow(1,-1)\rt{\int_{t_0}^{t_1}-\G(\sigma)\,d\sigma}
&&&&&&&&\arrow(1,1)\lft{\H(\ )}&\mapdown\rt
{\int_{t_0}^{t_1}-\H(\sigma)\,d\sigma}\cr
\Rn&\mapleft_{\W(t_0,t_1)}&\Rn&&&\olap{\mapright^{\I}}&&&
\Rn&\mapleft_{\M(t_0,t_1)}&\Rn\cr
\mapdown\lft{\P'}&&\mapup\lft\P&\arrow(-1,1)\rt\P&&&&\arrow(-1,-1)\lft\Q&
\mapdown\rt\Q&&\mapup\rt{\Q'}\cr
\Rn&\mapright^{\S_1}&\Rn&\mapright^{\S_1}&\Rn&\harrowlength=20pt
\mapright^{\Q\P}&\Rn&\mapright^{\S_2}&\Rn&\mapright^{\S_2}&\Rn\cr
&&\mapdown\lft{\N_1}&&&&&&\mapup\rt{\N_2}\cr
&&\Rn&&&\olap{\mapright^{\I}}&&&\Rn\cr}$$

Page 114, Fig.~1:
$$\sarrowlength=.42\harrowlength
\commdiag{&\Real^m\cr &\arrow(-1,-1)\lft\B\quad \arrow(1,-1)\rt\G\cr
\Rn&\mapright^\P&\Rn\cr \mapdown\lft{e^{\A t}}&&\mapdown\rt{e^{\F t}}\cr
\Rn&\mapright^\P&\Rn\cr &\arrow(1,-1)\lft\C\quad \arrow(-1,-1)\rt\H\cr
&\Real^q\cr}$$

Page 127, Fig.~2:
$$\commdiag{Y&\mapright^{L^*}&X\cr &\adjarrow(3,-2)\lft{LL^*}\rt{(LL^*)^{-1}}&
\mapdown\rt{L}\cr &&Y\cr}$$

Page 135, Fig.~2:
$$\harrowlength=80pt \varrowlength=40pt \sarrowlength=50pt
\def\olap#1{\harrowlength=214pt\hbox to0pt{\hss$#1$\hss}}
\def\vlap#1{\varrowlength=93pt\setbox0=\hbox{$#1$}\ht0=0pt\dp0=0pt\box0}
\commdiag{\Rn&&&\olap{\mapright^{\dot\K+\A'\K+\K\A+\L}}&&&\Rn\cr
&\arrow(3,-1)\rt\K&&&&\arrow(-3,-1)\lft\K\cr \noalign{\vskip-3pt}
&&\Rn&\mapright^{\B\B'}&\Rn\cr
\vlap{\mapdown\lft\P}&&\mapup\lft{\P'}&&\mapdown\rt\P&&\vlap{\mapup\rt{\P'}}\cr
&&\Rn&\mapleft^{\P\B\B'\P'}&\Rn\cr \noalign{\vskip-3pt}
&\arrow(3,1)\lft{\K_1}&&&&\arrow(-3,1)\rt{\K_1}\cr
\Rn&&&\olap{\mapright_{\dot\K_1+(\P\A\P^{-1}+\dot\P\P^{-1})\K_1
+\K_1(\P\A\P^{-1}+\dot\P\P^{-1})+\P'^{-1}\L\P^{-1}}}&&&\Rn\cr
\noalign{\medskip}}$$

Page 139, Fig.~2:
$$\harrowlength=80pt \varrowlength=.618\harrowlength
\sarrowlength=\harrowlength
\commdiag{\Rn&\mapright^{\B'(\sigma)\bp'(t_0,\sigma)}&C^m\cr
&\adjarrow(3,-2)\lft{\W(t_0,t_1)}\rt{\W^{-1}(t_0,t_1)}&
\mapdown\rt{\int_{t_0}^{t_1}\bp(t_0,\sigma)\B(\sigma)\,d\sigma}\cr
&&\Rn\cr}$$
