X-Git-Url: http://gitweb.michael.orlitzky.com/?p=mjotex.git;a=blobdiff_plain;f=mjo-linear_algebra.tex;h=d493b9cce5bd047eeb940a972128a6cf7198d261;hp=d423a433e65030774141eac013cf95bbea2a1aa4;hb=2398b156d5cec30d4ede3e65ae1c89ad08551447;hpb=fc7cfa7f3d02715cb09eae2e1c6bc501dc2d8d50

diff --git a/mjo-linear_algebra.tex b/mjo-linear_algebra.tex
index d423a43..d493b9c 100644
--- a/mjo-linear_algebra.tex
+++ b/mjo-linear_algebra.tex
@@ -5,16 +5,20 @@
 \def\havemjolinearalgebra{1}
 
 
-% Needed for \lvert, \rVert, etc. and \operatorname.
-\usepackage{amsmath}
+\ifx\lvert\undefined
+  \usepackage{amsmath} % \lvert, \rVert, etc. and \operatorname.
+\fi
 
-% Wasysym contains the \ocircle that we use in \directsumperp.
-\usepackage{wasysym}
+\ifx\ocircle\undefined
+  \usepackage{wasysym}
+\fi
 
-% Part of the adjustbox package; needed to clip the \perp sign.
-\usepackage{trimclip}
+\ifx\clipbox\undefined
+  % Part of the adjustbox package; needed to clip the \perp sign.
+  \usepackage{trimclip}
+\fi
 
-\input{mjo-common}
+\input{mjo-common} % for \of, at least
 
 % Absolute value (modulus) of a scalar.
 \newcommand*{\abs}[1]{\left\lvert{#1}\right\rvert}
@@ -47,6 +51,17 @@
 % 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}} }
 
@@ -56,6 +71,19 @@
 % The orthogonal projection of its second argument onto the first.
 \newcommand*{\proj}[2] { \operatorname{proj}\of{#1, #2} }
 
+% The set of all eigenvalues of its argument, which should be either a
+% matrix or a linear operator. The sigma notation was chosen instead
+% of lambda so that lambda can be reserved to denote the ordered tuple
+% (largest to smallest) of eigenvalues.
+\newcommand*{\spectrum}[1]{\sigma\of{{#1}}}
+\ifdefined\newglossaryentry
+  \newglossaryentry{spectrum}{
+    name={\ensuremath{\spectrum{L}}},
+    description={the set of all eigenvalues of $L$},
+    sort=s
+  }
+\fi
+
 % The ``Automorphism group of'' operator.
 \newcommand*{\Aut}[1]{ \operatorname{Aut}\of{{#1}} }
 
@@ -104,4 +132,29 @@
 \newcommand*{\directsumperp}[2]{ {#1}\oplusperp{#2} }
 
 
+% The space of real symmetric n-by-n matrices. Does not reduce to
+% merely "S" when n=1 since S^{n} does not mean an n-fold cartesian
+% product of S^{1}.
+\newcommand*{\Sn}[1][n]{ \mathcal{S}^{#1} }
+\ifdefined\newglossaryentry
+  \newglossaryentry{Sn}{
+    name={\ensuremath{\Sn}},
+    description={the set of $n$-by-$n$ real symmetric matrices},
+    sort=Sn
+  }
+\fi
+
+% The space of complex Hermitian n-by-n matrices. Does not reduce to
+% merely "H" when n=1 since H^{n} does not mean an n-fold cartesian
+% product of H^{1}.
+\newcommand*{\Hn}[1][n]{ \mathcal{H}^{#1} }
+\ifdefined\newglossaryentry
+  \newglossaryentry{Hn}{
+    name={\ensuremath{\Hn}},
+    description={the set of $n$-by-$n$ complex Hermitian matrices},
+    sort=Hn
+  }
+\fi
+
+
 \fi