mjo-cone.tex

   1 %
   2 % Cone stuff.
   3 %
   4 % The operator families Z(K), LL(K), etc. can technically be defined on
   5 % sets other than cones, but nobody cares.
   6 %
   7 \ifx\havemjocone\undefined
   8 \def\havemjocone{1}
   9
  10
  11 \ifx\succcurlyeq\undefined
  12   \usepackage{amssymb} % \succcurlyeq, \preccurlyeq
  13 \fi
  14
  15 \input{mjo-common} % for \of, \Rn, etc.
  16 \input{mjo-linear_algebra} % \Sn and \Hn
  17
  18 % The dual of a subset of an inner-product space; always a closed
  19 % convex cone.
  20 \newcommand*{\dual}[1]{ #1^{*} }
  21
  22 %
  23 % Common cones.
  24 %
  25
  26 % The nonnegative orthant in the given number of dimensions.
  27 \newcommand*{\Rnplus}[1][n]{ \Rn[#1]_{+} }
  28
  29 % The Lorentz ``ice-cream'' cone in the given number of dimensions.
  30 \newcommand*{\Lnplus}[1][n]{ \mathcal{L}^{{#1}}_{+} }
  31
  32 % The PSD cone in a space of symmetric matrices.
  33 \newcommand*{\Snplus}[1][n]{ \Sn[#1]_{+} }
  34
  35 % The PSD cone in a space of Hermitian matrices.
  36 \newcommand*{\Hnplus}[1][n]{ \Hn[#1]_{+} }
  37
  38
  39 %
  40 % Some collections of linear operators.
  41 %
  42
  43 % The set of all positive operators on its argument. This uses the
  44 % same magic as \boundedops to accept either one or two arguments. If
  45 % one argument is given, the domain and codomain are equal and the
  46 % positive operators fix a subset of that space. When two arguments
  47 % are given, the positive operators send the first argument to a
  48 % subset of the second.
  49 \newcommand*{\posops}[2][]{
  50   \pi\of{ {#2}
  51     \if\relax\detokenize{#1}\relax
  52       {}%
  53     \else
  54       {,{#1}}%
  55     \fi
  56   }
  57 }
  58
  59 % The set of all S-operators on its argument.
  60 \newcommand*{\Sof}[1]{ \mathbf{S} \of{ {#1} } }
  61
  62 % The cone of all Z-operators on its argument.
  63 \newcommand*{\Zof}[1]{ \mathbf{Z} \of{ {#1} } }
  64
  65 % The space of Lyapunov-like operators on its argument.
  66 \newcommand*{\LL}[1]{ \mathbf{LL}\of{ {#1} } }
  67
  68 % The Lyapunov rank of the given cone.
  69 \newcommand*{\lyapunovrank}[1]{ \beta\of{ {#1} } }
  70
  71 % Cone inequality operators.
  72 \newcommand*{\gecone}{\succcurlyeq}
  73 \newcommand*{\gtcone}{\succ}
  74 \newcommand*{\lecone}{\preccurlyeq}
  75 \newcommand*{\ltcone}{\prec}
  76
  77
  78 \fi