mjo-hurwitz.tex: new file for Hurwitz algebras (quaternions and octonions).
[mjotex.git] / 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