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]{\left\langle{#1},{#2}\right\rangle}
  18
  19 % The tensor product of its two arguments.
  20 \newcommand*{\tp}[2]{ {#1}\otimes{#2} }
  21
  22 % The Kronecker product of its two arguments. The usual notation for
  23 % this is the same as the tensor product notation used for \tp, but
  24 % that leads to confusion because the two definitions may not agree.
  25 \newcommand*{\kp}[2]{ {#1}\odot{#2} }
  26
  27 % The adjoint of a linear operator.
  28 \newcommand*{\adjoint}[1]{ #1^{*} }
  29
  30 % The ``transpose'' of a linear operator; namely, the adjoint, but
  31 % specialized to real matrices.
  32 \newcommand*{\transpose}[1]{ #1^{T} }
  33
  34 % The ``span of'' operator. The name \span is already taken.
  35 \newcommand*{\spanof}[1]{ \operatorname{span}\of{{#1}} }
  36
  37 % The ``co-dimension of'' operator.
  38 \newcommand*{\codim}{ \operatorname{codim} }
  39
  40 % The trace of an operator.
  41 \newcommand*{\trace}[1]{ \operatorname{trace}\of{{#1}} }
  42
  43 % The orthogonal projection of its second argument onto the first.
  44 \newcommand*{\proj}[2] { \operatorname{proj}\of{#1, #2} }
  45
  46 % The ``Automorphism group of'' operator.
  47 \newcommand*{\Aut}[1]{ \operatorname{Aut}\of{{#1}} }
  48
  49 % The ``Lie algebra of'' operator.
  50 \newcommand*{\Lie}[1]{ \operatorname{Lie}\of{{#1}} }
  51
  52 % The ``write a matrix as a big vector'' operator.
  53 \newcommand*{\vectorize}[1]{ \operatorname{vec}\of{{#1}} }
  54
  55 % The ``write a big vector as a matrix'' operator.
  56 \newcommand*{\matricize}[1]{ \operatorname{mat}\of{{#1}} }
  57
  58 % An inline column vector, with parentheses and a transpose operator.
  59 \newcommand*{\colvec}[1]{ \left({#1}\right)^{T} }
  60
  61 % Bounded linear operators on some space. The required argument is the
  62 % domain of those operators, and the optional argument is the
  63 % codomain. If the optional argument is omitted, the required argument
  64 % is used for both.
  65 \newcommand*{\boundedops}[2][]{
  66   \mathcal{B}\of{ {#2}
  67     \if\relax\detokenize{#1}\relax
  68       {}%
  69     \else
  70       {,{#1}}%
  71     \fi
  72   }
  73 }