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