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 % The diagonal matrix whose only nonzero entries are on the diagonal
  55 % and are given by our argument. The argument should therefore be a
  56 % vector or tuple of entries, by convention going from the top-left to
  57 % the bottom-right of the matrix.
  58 \newcommand*{\diag}[1]{\operatorname{diag}\of{{#1}}}
  59
  60 % The "rank" of its argument, which is context-dependent. It can mean
  61 % any or all of,
  62 %
  63 %   * the rank of a matrix,
  64 %   * the rank of a power-associative algebra (particularly an EJA),
  65 %   * the rank of an element in a Euclidean Jordan algebra.
  66 %
  67 \newcommand*{\rank}[1]{ \operatorname{rank}\of{{#1}} }
  68
  69
  70 % The ``span of'' operator. The name \span is already taken.
  71 \newcommand*{\spanof}[1]{ \operatorname{span}\of{{#1}} }
  72
  73 % The ``co-dimension of'' operator.
  74 \newcommand*{\codim}{ \operatorname{codim} }
  75
  76 % The orthogonal projection of its second argument onto the first.
  77 \newcommand*{\proj}[2] { \operatorname{proj}\of{#1, #2} }
  78
  79 % The set of all eigenvalues of its argument, which should be either a
  80 % matrix or a linear operator. The sigma notation was chosen instead
  81 % of lambda so that lambda can be reserved to denote the ordered tuple
  82 % (largest to smallest) of eigenvalues.
  83 \newcommand*{\spectrum}[1]{\sigma\of{{#1}}}
  84 \ifdefined\newglossaryentry
  85   \newglossaryentry{spectrum}{
  86     name={\ensuremath{\spectrum{L}}},
  87     description={the set of all eigenvalues of $L$},
  88     sort=s
  89   }
  90 \fi
  91
  92 % The ``Automorphism group of'' operator.
  93 \newcommand*{\Aut}[1]{ \operatorname{Aut}\of{{#1}} }
  94
  95 % The ``Lie algebra of'' operator.
  96 \newcommand*{\Lie}[1]{ \operatorname{Lie}\of{{#1}} }
  97
  98 % The ``write a matrix as a big vector'' operator.
  99 \newcommand*{\vectorize}[1]{ \operatorname{vec}\of{{#1}} }
 100
 101 % The ``write a big vector as a matrix'' operator.
 102 \newcommand*{\matricize}[1]{ \operatorname{mat}\of{{#1}} }
 103
 104 % An inline column vector, with parentheses and a transpose operator.
 105 \newcommand*{\colvec}[1]{ \transpose{\left({#1}\right)} }
 106
 107 % Bounded linear operators on some space. The required argument is the
 108 % domain of those operators, and the optional argument is the
 109 % codomain. If the optional argument is omitted, the required argument
 110 % is used for both.
 111 \newcommand*{\boundedops}[2][]{
 112   \mathcal{B}\of{ {#2}
 113     \if\relax\detokenize{#1}\relax
 114       {}%
 115     \else
 116       {,{#1}}%
 117     \fi
 118   }
 119 }
 120
 121
 122 %
 123 % Orthogonal direct sum.
 124 %
 125 % First declare my ``perp in a circle'' operator, which is meant to be
 126 % like an \obot or an \operp except has the correct weight circle. It's
 127 % achieved by overlaying an \ocircle with a \perp, but only after we
 128 % clip off the top half of the \perp sign and shift it up.
 129 \DeclareMathOperator{\oplusperp}{\mathbin{
 130   \ooalign{
 131     $\ocircle$\cr
 132     \raisebox{0.625\height}{$\clipbox{0pt 0pt 0pt 0.5\height}{$\perp$}$}\cr
 133   }
 134 }}
 135
 136 % Now declare an orthogonal direct sum in terms of \oplusperp.
 137 \newcommand*{\directsumperp}[2]{ {#1}\oplusperp{#2} }
 138
 139
 140 % The space of real symmetric n-by-n matrices. Does not reduce to
 141 % merely "S" when n=1 since S^{n} does not mean an n-fold cartesian
 142 % product of S^{1}.
 143 \newcommand*{\Sn}[1][n]{ \mathcal{S}^{#1} }
 144 \ifdefined\newglossaryentry
 145   \newglossaryentry{Sn}{
 146     name={\ensuremath{\Sn}},
 147     description={the set of $n$-by-$n$ real symmetric matrices},
 148     sort=Sn
 149   }
 150 \fi
 151
 152 % The space of complex Hermitian n-by-n matrices. Does not reduce to
 153 % merely "H" when n=1 since H^{n} does not mean an n-fold cartesian
 154 % product of H^{1}. The field may also be given rather than assumed
 155 % to be complex; for example \Hn[3]\of{\mathbb{O}} might denote the
 156 % 3-by-3 Hermitian matrices with octonion entries.
 157 \newcommand*{\Hn}[1][n]{ \mathcal{H}^{#1} }
 158 \ifdefined\newglossaryentry
 159   \newglossaryentry{Hn}{
 160     name={\ensuremath{\Hn}},
 161     description={the set of $n$-by-$n$ complex Hermitian matrices},
 162     sort=Hn
 163   }
 164 \fi
 165
 166
 167 \fi