mjo-common: prevent \Sn[1] and \Hn[1] from dropping their superscripts.

[mjotex.git] / mjo-linear_algebra.tex

diff --git a/mjo-linear_algebra.tex b/mjo-linear_algebra.tex
index e11a9458843f56484a08538111063f3de869c9f8..204ad05bd6b7981d32e2c56a4b268642d58d4ce2 100644 (file)
--- a/mjo-linear_algebra.tex
+++ b/mjo-linear_algebra.tex
@@ -1,33 +1,73 @@
 %
 % Standard operations from linear algebra.
 %
+\ifx\havemjolinearalgebra\undefined
+\def\havemjolinearalgebra{1}
 
-% Needed for \lvert, \rVert, etc. and \operatorname.
-\usepackage{amsmath}
+
+\ifx\lvert\undefined
+  \usepackage{amsmath} % \lvert, \rVert, etc. and \operatorname.
+\fi
+
+\ifx\ocircle\undefined
+  \usepackage{wasysym}
+\fi
+
+\ifx\clipbox\undefined
+  % Part of the adjustbox package; needed to clip the \perp sign.
+  \usepackage{trimclip}
+\fi
 
 \input{mjo-common}
 
-% Absolute value (modulis) of a scalar.
+% Absolute value (modulus) of a scalar.
 \newcommand*{\abs}[1]{\left\lvert{#1}\right\rvert}
 
 % Norm of a vector.
 \newcommand*{\norm}[1]{\left\lVert{#1}\right\rVert}
 
 % The inner product between its two arguments.
-\newcommand*{\ip}[2]{\langle{#1},{#2}\rangle}
+\newcommand*{\ip}[2]{\left\langle{#1},{#2}\right\rangle}
 
 % The tensor product of its two arguments.
 \newcommand*{\tp}[2]{ {#1}\otimes{#2} }
 
+% The Kronecker product of its two arguments. The usual notation for
+% this is the same as the tensor product notation used for \tp, but
+% that leads to confusion because the two definitions may not agree.
+\newcommand*{\kp}[2]{ {#1}\odot{#2} }
+
+% The adjoint of a linear operator.
+\newcommand*{\adjoint}[1]{ #1^{*} }
+
+% The ``transpose'' of a linear operator; namely, the adjoint, but
+% specialized to real matrices.
+\newcommand*{\transpose}[1]{ #1^{T} }
+
+% The Moore-Penrose (or any other, I guess) pseudo-inverse of its
+% sole argument.
+\newcommand*{\pseudoinverse}[1]{ #1^{+} }
+
+% The trace of an operator.
+\newcommand*{\trace}[1]{ \operatorname{trace}\of{{#1}} }
+
+
+% The "rank" of its argument, which is context-dependent. It can mean
+% any or all of,
+%
+%   * the rank of a matrix,
+%   * the rank of a power-associative algebra (particularly an EJA),
+%   * the rank of an element in a Euclidean Jordan algebra.
+%
+\newcommand*{\rank}[1]{ \operatorname{rank}\of{{#1}} }
+
+
 % The ``span of'' operator. The name \span is already taken.
 \newcommand*{\spanof}[1]{ \operatorname{span}\of{{#1}} }
 
 % The ``co-dimension of'' operator.
 \newcommand*{\codim}{ \operatorname{codim} }
 
-% The trace of an operator.
-\newcommand*{\trace}[1]{ \operatorname{trace}\of{{#1}} }
-
 % The orthogonal projection of its second argument onto the first.
 \newcommand*{\proj}[2] { \operatorname{proj}\of{#1, #2} }
 
@@ -44,7 +84,7 @@
 \newcommand*{\matricize}[1]{ \operatorname{mat}\of{{#1}} }
 
 % An inline column vector, with parentheses and a transpose operator.
-\newcommand*{\colvec}[1]{ \left({#1}\right)^{T} }
+\newcommand*{\colvec}[1]{ \transpose{\left({#1}\right)} }
 
 % Bounded linear operators on some space. The required argument is the
 % domain of those operators, and the optional argument is the
@@ -59,3 +99,24 @@
     \fi
   }
 }
+
+
+%
+% Orthogonal direct sum.
+%
+% First declare my ``perp in a circle'' operator, which is meant to be
+% like an \obot or an \operp except has the correct weight circle. It's
+% achieved by overlaying an \ocircle with a \perp, but only after we
+% clip off the top half of the \perp sign and shift it up.
+\DeclareMathOperator{\oplusperp}{\mathbin{
+  \ooalign{
+    $\ocircle$\cr
+    \raisebox{0.625\height}{$\clipbox{0pt 0pt 0pt 0.5\height}{$\perp$}$}\cr
+  }
+}}
+
+% Now declare an orthogonal direct sum in terms of \oplusperp.
+\newcommand*{\directsumperp}[2]{ {#1}\oplusperp{#2} }
+
+
+\fi