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

[mjotex.git] / examples.tex

diff --git a/examples.tex b/examples.tex
index f922d655715645da0e00309d1fb7973f7f4b3c51..0cf8afa51a90a84b624fee9a7f4232ae7ff9a52d 100644 (file)
--- a/examples.tex
+++ b/examples.tex
@@ -72,10 +72,10 @@
   \end{section}
 
   \begin{section}{Common}
-    The function $f$ applied to $x$ is $f\of{x}$. We can group terms
-    like $a + \qty{b - c}$ or $a + \qty{b - \sqty{c - d}}$. Here's a
-    set $\set{1,2,3} = \setc{n \in \Nn[1]}{ n \le 3 }$. The tuples go
-    up to seven, for now:
+    The function $f$ applied to $x$ is $f\of{x}$, and the restriction
+    of $f$ to a subset $X$ of its domain is $\restrict{f}{X}$. We can
+    group terms like $a + \qty{b - c}$ or $a + \qty{b - \sqty{c -
+        d}}$. The tuples go up to seven, for now:
     %
     \begin{itemize}
       \begin{item}
@@ -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-$n$
@@ -290,8 +291,9 @@
   \end{section}
 
   \begin{section}{Set theory}
-    The cardinality of the set $X \coloneqq \set{1,2,3}$ is $\card{X}
-    = 3$, and its powerset is $\powerset{X}$.
+    Here's a set $\set{1,2,3} = \setc{n \in \Nn[1]}{ n \le 3 }$. The
+    cardinality of the set $X \coloneqq \set{1,2,3}$ is $\card{X} =
+    3$, and its powerset is $\powerset{X}$.
 
     We also have a few basic set operations, for example the union of
     two or three sets: $\union{A}{B}$, $\unionthree{A}{B}{C}$. And of