X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo-linear_algebra.tex;h=71ab1b61362be810e0a758490641b86a7ae3f52f;hb=7c7ef6216956f7a5c8bbaf1d9780d2e1aa431342;hp=9e32d2179a25e1d061eeeb1f063ed14bcdf36284;hpb=87aa4249a1cb51b6393e7d6f74276f00efc15df0;p=mjotex.git

diff --git a/mjo-linear_algebra.tex b/mjo-linear_algebra.tex
index 9e32d21..71ab1b6 100644
--- a/mjo-linear_algebra.tex
+++ b/mjo-linear_algebra.tex
@@ -8,43 +8,51 @@
 \input{mjo-common}
 
 % Absolute value (modulis) of a scalar.
-\newcommand*{\abs}[1]{ \left\lvert {#1} \right\rvert }
+\newcommand*{\abs}[1]{\left\lvert{#1}\right\rvert}
 
 % Norm of a vector.
-\newcommand*{\norm}[1]{ \left\lVert {#1} \right\rVert }
+\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]{\langle{#1},{#2}\rangle}
 
 % The tensor product of its two arguments.
-\newcommand*{\tp}[2]{ {#1} \otimes {#2} }
+\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 ``span of'' operator. The name \span is already taken.
-\newcommand*{\spanof}[1]{ \operatorname{span} \of{{#1}} }
+\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}} }
+\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} }
 
 % The ``Automorphism group of'' operator.
-\newcommand*{\Aut}[1]{ \operatorname{Aut} \of{{#1}} }
+\newcommand*{\Aut}[1]{ \operatorname{Aut}\of{{#1}} }
 
 % The ``Lie algebra of'' operator.
-\newcommand*{\Lie}[1]{ \operatorname{Lie} \of{{#1}} }
+\newcommand*{\Lie}[1]{ \operatorname{Lie}\of{{#1}} }
 
 % The ``write a matrix as a big vector'' operator.
-\newcommand*{\vectorize}[1]{ \operatorname{vec} \of{{#1}} }
+\newcommand*{\vectorize}[1]{ \operatorname{vec}\of{{#1}} }
 
 % The ``write a big vector as a matrix'' operator.
-\newcommand*{\matricize}[1]{ \operatorname{mat} \of{{#1}} }
+\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]{ \left({#1}\right)^{T} }
 
 % Bounded linear operators on some space. The required argument is the
 % domain of those operators, and the optional argument is the