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