mjo-linear_algebra.tex

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