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 % The ``span of'' operator. The name \span is already taken.
  55 \newcommand*{\spanof}[1]{ \operatorname{span}\of{{#1}} }
  56
  57 % The ``co-dimension of'' operator.
  58 \newcommand*{\codim}{ \operatorname{codim} }
  59
  60 % The orthogonal projection of its second argument onto the first.
  61 \newcommand*{\proj}[2] { \operatorname{proj}\of{#1, #2} }
  62
  63 % The ``Automorphism group of'' operator.
  64 \newcommand*{\Aut}[1]{ \operatorname{Aut}\of{{#1}} }
  65
  66 % The ``Lie algebra of'' operator.
  67 \newcommand*{\Lie}[1]{ \operatorname{Lie}\of{{#1}} }
  68
  69 % The ``write a matrix as a big vector'' operator.
  70 \newcommand*{\vectorize}[1]{ \operatorname{vec}\of{{#1}} }
  71
  72 % The ``write a big vector as a matrix'' operator.
  73 \newcommand*{\matricize}[1]{ \operatorname{mat}\of{{#1}} }
  74
  75 % An inline column vector, with parentheses and a transpose operator.
  76 \newcommand*{\colvec}[1]{ \transpose{\left({#1}\right)} }
  77
  78 % Bounded linear operators on some space. The required argument is the
  79 % domain of those operators, and the optional argument is the
  80 % codomain. If the optional argument is omitted, the required argument
  81 % is used for both.
  82 \newcommand*{\boundedops}[2][]{
  83   \mathcal{B}\of{ {#2}
  84     \if\relax\detokenize{#1}\relax
  85       {}%
  86     \else
  87       {,{#1}}%
  88     \fi
  89   }
  90 }
  91
  92
  93 %
  94 % Orthogonal direct sum.
  95 %
  96 % First declare my ``perp in a circle'' operator, which is meant to be
  97 % like an \obot or an \operp except has the correct weight circle. It's
  98 % achieved by overlaying an \ocircle with a \perp, but only after we
  99 % clip off the top half of the \perp sign and shift it up.
 100 \DeclareMathOperator{\oplusperp}{\mathbin{
 101   \ooalign{
 102     $\ocircle$\cr
 103     \raisebox{0.625\height}{$\clipbox{0pt 0pt 0pt 0.5\height}{$\perp$}$}\cr
 104   }
 105 }}
 106
 107 % Now declare an orthogonal direct sum in terms of \oplusperp.
 108 \newcommand*{\directsumperp}[2]{ {#1}\oplusperp{#2} }
 109
 110
 111 \fi