mjo-linear_algebra.tex

   1 %
   2 % Standard operations from linear algebra.
   3 %
   4
   5 % Needed for \lvert, \rVert, etc. and \operatorname.
   6 \usepackage{amsmath}
   7
   8 \input{mjo-common}
   9
  10 % Absolute value (modulis) of a scalar.
  11 \newcommand*{\abs}[1]{ \left\lvert {#1} \right\rvert }
  12
  13 % Norm of a vector.
  14 \newcommand*{\norm}[1]{ \left\lVert {#1} \right\rVert }
  15
  16 % The inner product between its two arguments.
  17 \newcommand*{\ip}[2]{ \langle {#1}, {#2} \rangle }
  18
  19 % The tensor product of its two arguments.
  20 \newcommand*{\tp}[2]{ {#1} \otimes {#2} }
  21
  22 % The ``span of'' operator. The name \span is already taken.
  23 \newcommand*{\spanof}[1]{ \operatorname{span} \of{{#1}} }
  24
  25 % The ``co-dimension of'' operator.
  26 \newcommand*{\codim}{ \operatorname{codim} }
  27
  28 % The trace of an operator.
  29 \newcommand*{\trace}[1]{ \operatorname{trace} \of{{#1}} }
  30
  31 % The orthogonal projection of its second argument onto the first.
  32 \newcommand*{\proj}[2] { \operatorname{proj}\of{#1, #2} }
  33
  34 % The ``Automorphism group of'' operator.
  35 \newcommand*{\Aut}[1]{ \operatorname{Aut} \of{{#1}} }
  36
  37 % The ``Lie algebra of'' operator.
  38 \newcommand*{\Lie}[1]{ \operatorname{Lie} \of{{#1}} }
  39
  40 % The ``write a matrix as a big vector'' operator.
  41 \newcommand*{\vectorize}[1]{ \operatorname{vec} \of{{#1}} }
  42
  43 % The ``write a big vector as a matrix'' operator.
  44 \newcommand*{\matricize}[1]{ \operatorname{mat} \of{{#1}} }
  45
  46 % An inline column vector, with parentheses and a transpose operator.
  47 \newcommand*{\colvec}[1]{ \left( {#1} \right)^{T} }
  48
  49 % Bounded linear operators on some space. The required argument is the
  50 % domain of those operators, and the optional argument is the
  51 % codomain. If the optional argument is omitted, the required argument
  52 % is used for both.
  53 \newcommand*{\boundedops}[2][]{
  54   \mathcal{B}\of{ {#2}
  55     \if\relax\detokenize{#1}\relax
  56       {}%
  57     \else
  58       {,{#1}}%
  59     \fi
  60   }
  61 }