X-Git-Url: http://gitweb.michael.orlitzky.com/?p=mjotex.git;a=blobdiff_plain;f=mjo-linear_algebra.tex;h=a534523765ef1fd20f7729af25836bb2075fcd3e;hp=204ad05bd6b7981d32e2c56a4b268642d58d4ce2;hb=66a672c6c1657ad96422fcdb4828bc5d9a108ab0;hpb=8f3a3f952b0e5bd692b9b41d1417c877b0e0425e

diff --git a/mjo-linear_algebra.tex b/mjo-linear_algebra.tex
index 204ad05..a534523 100644
--- a/mjo-linear_algebra.tex
+++ b/mjo-linear_algebra.tex
@@ -18,7 +18,7 @@
   \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}
@@ -51,6 +51,11 @@
 % The trace of an operator.
 \newcommand*{\trace}[1]{ \operatorname{trace}\of{{#1}} }
 
+% The diagonal matrix whose only nonzero entries are on the diagonal
+% and are given by our argument. The argument should therefore be a
+% vector or tuple of entries, by convention going from the top-left to
+% the bottom-right of the matrix.
+\newcommand*{\diag}[1]{\operatorname{diag}\of{{#1}}}
 
 % The "rank" of its argument, which is context-dependent. It can mean
 % any or all of,
@@ -71,6 +76,19 @@
 % 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}} }
 
@@ -119,4 +137,31 @@
 \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}. The field may also be given rather than assumed
+% to be complex; for example \Hn[3]\of{\mathbb{O}} might denote the
+% 3-by-3 Hermitian matrices with octonion entries.
+\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