mjo-linear_algebra.tex

   1 %
   2 % Standard operations from linear algebra.
   3 %
   4 \ifx\havemjolinearalgebra\undefined
   5 \def\havemjolinearalgebra{1}
   6
   7
   8 \ifx\lvert\undefined
   9   \usepackage{amsmath} % \lvert, \rVert, etc. and \operatorname.
  10 \fi
  11
  12 \ifx\ocircle\undefined
  13   \usepackage{wasysym}
  14 \fi
  15
  16 \ifx\clipbox\undefined
  17   % Part of the adjustbox package; needed to clip the \perp sign.
  18   \usepackage{trimclip}
  19 \fi
  20
  21 \input{mjo-common}
  22
  23 % Absolute value (modulus) of a scalar.
  24 \newcommand*{\abs}[1]{\left\lvert{#1}\right\rvert}
  25
  26 % Norm of a vector.
  27 \newcommand*{\norm}[1]{\left\lVert{#1}\right\rVert}
  28
  29 % The inner product between its two arguments.
  30 \newcommand*{\ip}[2]{\left\langle{#1},{#2}\right\rangle}
  31
  32 % The tensor product of its two arguments.
  33 \newcommand*{\tp}[2]{ {#1}\otimes{#2} }
  34
  35 % The Kronecker product of its two arguments. The usual notation for
  36 % this is the same as the tensor product notation used for \tp, but
  37 % that leads to confusion because the two definitions may not agree.
  38 \newcommand*{\kp}[2]{ {#1}\odot{#2} }
  39
  40 % The adjoint of a linear operator.
  41 \newcommand*{\adjoint}[1]{ #1^{*} }
  42
  43 % The ``transpose'' of a linear operator; namely, the adjoint, but
  44 % specialized to real matrices.
  45 \newcommand*{\transpose}[1]{ #1^{T} }
  46
  47 % The Moore-Penrose (or any other, I guess) pseudo-inverse of its
  48 % sole argument.
  49 \newcommand*{\pseudoinverse}[1]{ #1^{+} }
  50
  51 % The trace of an operator.
  52 \newcommand*{\trace}[1]{ \operatorname{trace}\of{{#1}} }
  53
  54
  55 % The "rank" of its argument, which is context-dependent. It can mean
  56 % any or all of,
  57 %
  58 %   * the rank of a matrix,
  59 %   * the rank of a power-associative algebra (particularly an EJA),
  60 %   * the rank of an element in a Euclidean Jordan algebra.
  61 %
  62 \newcommand*{\rank}[1]{ \operatorname{rank}\of{{#1}} }
  63
  64
  65 % The ``span of'' operator. The name \span is already taken.
  66 \newcommand*{\spanof}[1]{ \operatorname{span}\of{{#1}} }
  67
  68 % The ``co-dimension of'' operator.
  69 \newcommand*{\codim}{ \operatorname{codim} }
  70
  71 % The orthogonal projection of its second argument onto the first.
  72 \newcommand*{\proj}[2] { \operatorname{proj}\of{#1, #2} }
  73
  74 % The ``Automorphism group of'' operator.
  75 \newcommand*{\Aut}[1]{ \operatorname{Aut}\of{{#1}} }
  76
  77 % The ``Lie algebra of'' operator.
  78 \newcommand*{\Lie}[1]{ \operatorname{Lie}\of{{#1}} }
  79
  80 % The ``write a matrix as a big vector'' operator.
  81 \newcommand*{\vectorize}[1]{ \operatorname{vec}\of{{#1}} }
  82
  83 % The ``write a big vector as a matrix'' operator.
  84 \newcommand*{\matricize}[1]{ \operatorname{mat}\of{{#1}} }
  85
  86 % An inline column vector, with parentheses and a transpose operator.
  87 \newcommand*{\colvec}[1]{ \transpose{\left({#1}\right)} }
  88
  89 % Bounded linear operators on some space. The required argument is the
  90 % domain of those operators, and the optional argument is the
  91 % codomain. If the optional argument is omitted, the required argument
  92 % is used for both.
  93 \newcommand*{\boundedops}[2][]{
  94   \mathcal{B}\of{ {#2}
  95     \if\relax\detokenize{#1}\relax
  96       {}%
  97     \else
  98       {,{#1}}%
  99     \fi
 100   }
 101 }
 102
 103
 104 %
 105 % Orthogonal direct sum.
 106 %
 107 % First declare my ``perp in a circle'' operator, which is meant to be
 108 % like an \obot or an \operp except has the correct weight circle. It's
 109 % achieved by overlaying an \ocircle with a \perp, but only after we
 110 % clip off the top half of the \perp sign and shift it up.
 111 \DeclareMathOperator{\oplusperp}{\mathbin{
 112   \ooalign{
 113     $\ocircle$\cr
 114     \raisebox{0.625\height}{$\clipbox{0pt 0pt 0pt 0.5\height}{$\perp$}$}\cr
 115   }
 116 }}
 117
 118 % Now declare an orthogonal direct sum in terms of \oplusperp.
 119 \newcommand*{\directsumperp}[2]{ {#1}\oplusperp{#2} }
 120
 121
 122 \fi