From: Michael Orlitzky <michael@orlitzky.com>
Date: Thu, 21 Nov 2019 19:28:24 +0000 (-0500)
Subject: mjo-linear_algebra: add the \spectrum{} of a linear operator.
X-Git-Url: https://gitweb.michael.orlitzky.com/?a=commitdiff_plain;h=6dfb93ac68463f1f47a009e2d1672c1c78f1e847;p=mjotex.git

mjo-linear_algebra: add the \spectrum{} of a linear operator.
---

diff --git a/examples.tex b/examples.tex
index c795973..0cf8afa 100644
--- a/examples.tex
+++ b/examples.tex
@@ -170,7 +170,8 @@
     their tensor product is $\tp{x}{y}$. The Kronecker product of
     matrices $A$ and $B$ is $\kp{A}{B}$. The adjoint of the operator
     $L$ is $\adjoint{L}$, or if it's a matrix, then its transpose is
-    $\transpose{L}$. Its trace is $\trace{L}$. Another matrix-specific
+    $\transpose{L}$. Its trace is $\trace{L}$, and its spectrum---the
+    set of its eigenvalues---is $\spectrum{L}$. Another matrix-specific
     concept is the Moore-Penrose pseudoinverse of $L$, denoted by
     $\pseudoinverse{L}$. Finally, the rank of a matrix $L$ is
     $\rank{L}$. As far as matrix spaces go, we have the $n$-by-$n$
diff --git a/mjo-linear_algebra.tex b/mjo-linear_algebra.tex
index c1aa9f5..d493b9c 100644
--- a/mjo-linear_algebra.tex
+++ b/mjo-linear_algebra.tex
@@ -71,6 +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}} }