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gitweb.michael.orlitzky.com - mjotex.git/blob - mjo-cone.tex
4 % The operator families Z(K), LL(K), etc. can technically be defined on
5 % sets other than cones, but nobody cares.
7 \ifx\havemjocone\undefined
11 \ifx\succcurlyeq\undefined
12 \usepackage{amssymb
} % \succcurlyeq, \preccurlyeq
15 \input{mjo-common
} % for \of, \Rn, etc.
16 \input{mjo-linear_algebra
} % \Sn and \Hn
18 % The dual of a subset of an inner-product space; always a closed
20 \newcommand*
{\dual}[1]{ #1^
{*
} }
26 % The nonnegative and strictly positive orthants in the given number
28 \newcommand*
{\Rnplus}[1][n
]{ \Rn[#1]_
{+
} }
29 \newcommand*
{\Rnplusplus}[1][n
]{ \Rn[#1]_
{++
} }
31 % The Lorentz ``ice-cream'' cone in the given number of dimensions.
32 \newcommand*
{\Lnplus}[1][n
]{ \mathcal{L
}^
{{#1}}_
{+
} }
34 % The PSD cone in a space of symmetric matrices.
35 \newcommand*
{\Snplus}[1][n
]{ \Sn[#1]_
{+
} }
37 % The PSD cone in a space of Hermitian matrices.
38 \newcommand*
{\Hnplus}[1][n
]{ \Hn[#1]_
{+
} }
42 % Some collections of linear operators.
45 % The set of all positive operators on its argument. This uses the
46 % same magic as \boundedops to accept either one or two arguments. If
47 % one argument is given, the domain and codomain are equal and the
48 % positive operators fix a subset of that space. When two arguments
49 % are given, the positive operators send the first argument to a
50 % subset of the second.
51 \newcommand*
{\posops}[2][]{
53 \if\relax\detokenize{#1}\relax
61 % The set of all S-operators on its argument.
62 \newcommand*
{\Sof}[1]{ \mathbf{S
} \of{ {#1} } }
64 % The cone of all Z-operators on its argument.
65 \newcommand*
{\Zof}[1]{ \mathbf{Z
} \of{ {#1} } }
67 % The space of Lyapunov-like operators on its argument.
68 \newcommand*
{\LL}[1]{ \mathbf{LL
}\of{ {#1} } }
70 % The Lyapunov rank of the given cone.
71 \newcommand*
{\lyapunovrank}[1]{ \beta\of{ {#1} } }
73 % Cone inequality operators.
74 \newcommand*
{\gecone}{\succcurlyeq}
75 \newcommand*
{\gtcone}{\succ}
76 \newcommand*
{\lecone}{\preccurlyeq}
77 \newcommand*
{\ltcone}{\prec}