X-Git-Url: http://gitweb.michael.orlitzky.com/?p=mjotex.git;a=blobdiff_plain;f=mjo-linear_algebra.tex;h=048022935b681bfe5f0d0c04366f7fadcb6383d2;hp=380c3bf546317349b20adedd1974bb82da6ed9f6;hb=5c4f67545f0988d065f1d52f90eed9233562c9fc;hpb=21d18c2a88952e968b02c838abc5cb61dff960c9

diff --git a/mjo-linear_algebra.tex b/mjo-linear_algebra.tex
index 380c3bf..0480229 100644
--- a/mjo-linear_algebra.tex
+++ b/mjo-linear_algebra.tex
@@ -5,16 +5,22 @@
 % Needed for \lvert, \rVert, etc. and \operatorname.
 \usepackage{amsmath}
 
+% Wasysym contains the \ocircle that we use in \directsumperp.
+\usepackage{wasysym}
+
+% Part of the adjustbox package; needed to clip the \perp sign.
+\usepackage{trimclip}
+
 \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} }
@@ -27,15 +33,19 @@
 % 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 trace of an operator.
+\newcommand*{\trace}[1]{ \operatorname{trace}\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} }
 
@@ -67,3 +77,21 @@
     \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{0.65\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} }