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} % for \of, at least
  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 set of all eigenvalues of its argument, which should be either a
  75 % matrix or a linear operator. The sigma notation was chosen instead
  76 % of lambda so that lambda can be reserved to denote the ordered tuple
  77 % (largest to smallest) of eigenvalues.
  78 \newcommand*{\spectrum}[1]{\sigma\of{{#1}}}
  79 \ifdefined\newglossaryentry
  80   \newglossaryentry{spectrum}{
  81     name={\ensuremath{\spectrum{L}}},
  82     description={the set of all eigenvalues of $L$},
  83     sort=s
  84   }
  85 \fi
  86
  87 % The ``Automorphism group of'' operator.
  88 \newcommand*{\Aut}[1]{ \operatorname{Aut}\of{{#1}} }
  89
  90 % The ``Lie algebra of'' operator.
  91 \newcommand*{\Lie}[1]{ \operatorname{Lie}\of{{#1}} }
  92
  93 % The ``write a matrix as a big vector'' operator.
  94 \newcommand*{\vectorize}[1]{ \operatorname{vec}\of{{#1}} }
  95
  96 % The ``write a big vector as a matrix'' operator.
  97 \newcommand*{\matricize}[1]{ \operatorname{mat}\of{{#1}} }
  98
  99 % An inline column vector, with parentheses and a transpose operator.
 100 \newcommand*{\colvec}[1]{ \transpose{\left({#1}\right)} }
 101
 102 % Bounded linear operators on some space. The required argument is the
 103 % domain of those operators, and the optional argument is the
 104 % codomain. If the optional argument is omitted, the required argument
 105 % is used for both.
 106 \newcommand*{\boundedops}[2][]{
 107   \mathcal{B}\of{ {#2}
 108     \if\relax\detokenize{#1}\relax
 109       {}%
 110     \else
 111       {,{#1}}%
 112     \fi
 113   }
 114 }
 115
 116
 117 %
 118 % Orthogonal direct sum.
 119 %
 120 % First declare my ``perp in a circle'' operator, which is meant to be
 121 % like an \obot or an \operp except has the correct weight circle. It's
 122 % achieved by overlaying an \ocircle with a \perp, but only after we
 123 % clip off the top half of the \perp sign and shift it up.
 124 \DeclareMathOperator{\oplusperp}{\mathbin{
 125   \ooalign{
 126     $\ocircle$\cr
 127     \raisebox{0.625\height}{$\clipbox{0pt 0pt 0pt 0.5\height}{$\perp$}$}\cr
 128   }
 129 }}
 130
 131 % Now declare an orthogonal direct sum in terms of \oplusperp.
 132 \newcommand*{\directsumperp}[2]{ {#1}\oplusperp{#2} }
 133
 134
 135 % The space of real symmetric n-by-n matrices. Does not reduce to
 136 % merely "S" when n=1 since S^{n} does not mean an n-fold cartesian
 137 % product of S^{1}.
 138 \newcommand*{\Sn}[1][n]{ \mathcal{S}^{#1} }
 139 \ifdefined\newglossaryentry
 140   \newglossaryentry{Sn}{
 141     name={\ensuremath{\Sn}},
 142     description={the set of $n$-by-$n$ real symmetric matrices},
 143     sort=Sn
 144   }
 145 \fi
 146
 147 % The space of complex Hermitian n-by-n matrices. Does not reduce to
 148 % merely "H" when n=1 since H^{n} does not mean an n-fold cartesian
 149 % product of H^{1}.
 150 \newcommand*{\Hn}[1][n]{ \mathcal{H}^{#1} }
 151 \ifdefined\newglossaryentry
 152   \newglossaryentry{Hn}{
 153     name={\ensuremath{\Hn}},
 154     description={the set of $n$-by-$n$ complex Hermitian matrices},
 155     sort=Hn
 156   }
 157 \fi
 158
 159
 160 \fi