{VERSION 3 0 "IBM INTEL NT" "3.0" }
{USTYLETAB {CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 
0 }{CSTYLE "2D Input" 2 19 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 }
{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "
Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 
0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" 
-1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 
0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 
0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }}
{SECT 0 {EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "with(linalg):" "6#-%%w
ithG6#%'linalgG" }}}{PARA 0 "" 0 "" {TEXT -1 242 "Repeatedly position \+
the cursor on the lines below that define matrices and hit return twic
e.  What observations do you make or what patterns do you find in the \+
associated eigenvalues and eigenvectors?  Which of your conjectures ca
n you prove?" }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "a2 := randmatr
ix(2,2);" "6#>%#a2G-%+randmatrixG6$\"\"#\"\"#" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#a2G-%'matrixG6#7$7$\"\"%!#f7$\"#i!#b" }}}{EXCHG 
{PARA 0 "> " 0 "" {XPPEDIT 19 1 "eva2 := eigenvects(a2);" "6#>%%eva2G-
%+eigenvectsG6#%#a2G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$7%,&#!#^\"\"#
\"\"\"*&%\"IGF(-%%sqrtG6#\"%R7\"\"\"#\"\"$F'F(<#-%'vectorG6#7$F(,&#F(F
'F(F)#!\"$\"$=\"7%,&F%F(F)#F:F'F(<#-F46#7$F(,&F8F(F)#F1F;" }}}{EXCHG 
{PARA 0 "> " 0 "" {XPPEDIT 19 1 "b2 := randmatrix(2,2,symmetric);" "6#
>%#b2G-%+randmatrixG6%\"\"#\"\"#%*symmetricG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#b2G-%'matrixG6#7$7$!#**!#n7$F+\"#o" }}}{EXCHG {PARA 
0 "> " 0 "" {XPPEDIT 19 1 "evb2 := eigenvects(b2);" "6#>%%evb2G-%+eige
nvectsG6#%#b2G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%evb2G6$7%,&#!#J\"
\"#\"\"\"*$-%%sqrtG6#\"&Xe%\"\"\"#F+F*F+<#-%'vectorG6#7$,&#\"$n\"\"$M
\"F+F,#!\"\"F;F+7%,&F(F+F,#F=F*F+<#-F56#7$,&F9F+F,#F+F;F+" }}}{PARA 0 
"" 0 "" {TEXT -1 103 "The following command demonstrates how to isolat
e an eigenvector so you can use it in further commands." }{TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "op(evb2[1,3]);" "6#-%#opG6
#&%%evb2G6$\"\"\"\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#
7$,&#\"$n\"\"$M\"\"\"\"*$-%%sqrtG6#\"&Xe%\"\"\"#!\"\"F*F+" }}}{EXCHG 
{PARA 0 "> " 0 "" {XPPEDIT 19 1 "%?;" "6#%#%?G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%#%?G" }}}{PARA 0 "" 0 "" {TEXT -1 89 "Make a conjectur
e about the nature of eigenvalues and eigenvectors of symmetric matric
es." }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "c2 := randmatrix(2,2,an
tisymmetric);" "6#>%#c2G-%+randmatrixG6%\"\"#\"\"#%.antisymmetricG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c2G-%'matrixG6#7$7$\"\"!\"#i7$!#iF*
" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "evc2 := eigenvects(c2);" 
"6#>%%evc2G-%+eigenvectsG6#%#c2G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$7%
,$%\"IG\"#i\"\"\"<#-%'vectorG6#7$,$F%!\"\"F'7%,$F%!#iF'<#-F*6#7$F%F'" 
}}}{PARA 0 "" 0 "" {TEXT -1 104 "Make a conjecture about the nature of
 eigenvalues and eigenvectors of anti (or skew) symmetric matrices." }
}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "d2 := randmatrix(2,2,entries \+
= rand(-9 .. 9));" "6#>%#d2G-%+randmatrixG6%\"\"#\"\"#/%(entriesG-%%ra
ndG6#;,$\"\"*!\"\"\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#d2G-%'ma
trixG6#7$7$\"\")\"\"!7$F*!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 
19 1 "QRdecomp(d2,Q = ('q2'));" "6#-%)QRdecompG6$%#d2G/%\"QG.%#q2G" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$,$*$-%%sqrtG6#\"\"#\"
\"\"\"\"),$F)!\"\"7$\"\"!F)" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 
1 "evalm(q2);" "6#-%&evalmG6#%#q2G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
-%'matrixG6#7$7$,$*$-%%sqrtG6#\"\"#\"\"\"#\"\"\"F-F(7$F(,$F)#!\"\"F-" 
}}}{PARA 0 "" 0 "" {TEXT -1 37 "q2 is a random 2x2 orthogonal matrix.
" }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "evq2 := eigenvects(q2);" "
6#>%%evq2G-%+eigenvectsG6#%#q2G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%
evq2G6$7%!\"\"\"\"\"<#-%'vectorG6#7$,&*$-%%sqrtG6#\"\"#\"\"\"F'F(F(F(7
%F(F(<#-F+6#7$,&F/F(F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 90 "Make a con
jecture about the nature of eigenvalues and eigenvectors of orthogonal
 matrices." }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "a3 := randmatrix
(3,3);" "6#>%#a3G-%+randmatrixG6$\"\"$\"\"$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#a3G-%'matrixG6#7%7%\"#\")!\"&!#G7%\"\"%!#6\"#57%\"#d
!##)!#[" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "eva3 := evalf(eige
nvects(a3));" "6#>%%eva3G-%&evalfG6#-%+eigenvectsG6#%#a3G" }}{PARA 12 
"" 1 "" {XPPMATH 20 "6%7%$\"+5%pP)o!\")$\"\"\"\"\"!<#-%'vectorG6#7%$\"
+e?m#y*!\"*F'$\"+)e/22%F17%,&$!+1Z)=M#F&F(%\"IG$\"+M^'Q$HF&F'<#-F,6#7%
,&$!+/)oJ6%!#5F(F8$\"*jf81'F1F',&$!+`zNx5F1F(F8$\"+\\2T\"p#F17%,&F6F(F
8$!+M^'Q$HF&F'<#-F,6#7%,&F@F(F8$!*jf81'F1F',&FFF(F8$!+\\2T\"p#F1" }}}
{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "b3 := randmatrix(3,3,symmetric
);" "6#>%#b3G-%+randmatrixG6%\"\"$\"\"$%*symmetricG" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%#b3G-%'matrixG6#7%7%!#6\"#Q\"#e7%F+!\"(!#%*7%F,F/!#
o" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "evb3 := evalf(eigenvects
(b3));" "6#>%%evb3G-%&evalfG6#-%+eigenvectsG6#%#b3G" }}{PARA 12 "" 1 "
" {XPPMATH 20 "6%7%,&$\"+J8%e:'!\")\"\"\"%\"IG$\"\"$F'$F(\"\"!<#-%'vec
torG6#7%,&$!))R'f()!\"*F(F)$!\"$F6F,,&$!+C8cZw!#5F(F)$!\"#F67%,&$!+Ffo
q;!\"(F(F)$F(F'F,<#-F06#7%,&$!)qg=wF'F(F)$F(F6F,,&$\"+H6([R\"F6F(F)$F+
F<7%,&$\"+Lz,^>F'F(F)$F8F'F,<#-F06#7%,&$\"+h*oS1$F6F(F)$\"\"#F6F,,&$\"
+;*Hkc*F<F(F)$\"\"*F<" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "c3 :
= randmatrix(3,3,antisymmetric);" "6#>%#c3G-%+randmatrixG6%\"\"$\"\"$%
.antisymmetricG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c3G-%'matrixG6#7
%7%\"\"!\"#!)\"#s7%!#!)F*\"#m7%!#s!#mF*" }}}{EXCHG {PARA 0 "> " 0 "" 
{XPPEDIT 19 1 "evc3 = eigenvects(c3);" "6#/%%evc3G-%+eigenvectsG6#%#c3
G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6%7%,$*&%\"IG\"\"\"-%%sqrtG6#\"%&)R
\"\"\"\"\"#F'<#-%'vectorG6#7%F',&#\"$(H\"$C(F'F%#\"\"&\"$i$,&#!$l\"F9F
'F%#\"\"*F67%,$F%!\"#F'<#-F06#7%F',&F4F'F%#!\"&F9,&F;F'F%#!\"*F67%\"\"
!F'<#-F06#7%F'#!#7\"#6#\"#S\"#L" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 104 "The following procedure generates random
 complex numbers whose real parts are integers between -9 and 9." }}
{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "f := proc () local z; z := ran
d(-9 .. 9)()+I*rand(-9 .. 9)() end;" "6#>%\"fGR6\"7#%\"zGF&F&>F(,&--%%
randG6#;,$\"\"*!\"\"\"\"*F&\"\"\"*&%\"IGF4--F-6#;,$\"\"*F2\"\"*F&F4F4F
&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6\"6#%\"zGF&F&>8$,&--%
%randG6#;!\"*\"\"*F&\"\"\"*&%\"IGF3F,F3F3F&F&F&" }}}{EXCHG {PARA 0 "> \+
" 0 "" {XPPEDIT 19 1 "f();" "6#-%\"fG6\"" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,&!\"\"\"\"\"%\"IG!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 
1 "m := randmatrix(2,2,entries = f);" "6#>%\"mG-%+randmatrixG6%\"\"#\"
\"#/%(entriesG%\"fG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG-%'matrix
G6#7$7$,&!\"$\"\"\"%\"IG\"\"&,&F.F,F-F.7$,&!\"\"F,F-!\"%,&\"\"%F,F-!\"
)" }}}{PARA 0 "" 0 "" {TEXT -1 52 "What property does the following ma
trix always have?" }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "hm := eva
lm(`&*`(htranspose(m),m));" "6#>%#hmG-%&evalmG6#-%#&*G6$-%+htransposeG
6#%\"mGF." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#hmG-%'matrixG6#7$7$\"#
^,&\"#Q\"\"\"%\"IG!#;7$,&F,F-F.\"#;\"$I\"" }}}{EXCHG {PARA 0 "> " 0 "
" {XPPEDIT 19 1 "evhm := eigenvects(hm);" "6#>%%evhmG-%+eigenvectsG6#%
#hmG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6$7%,&#\"$\"=\"\"#\"\"\"*$-%%sqr
tG6#\"$h\"\"\"\"#\"\"*F'F(<#-%'vectorG6#7$F(,*#\"%,:\"%+<F(F)#\"$r\"F9
*&%\"IGF(F$F(#\"\"%\"$D%F=#!#7\"#D7%,&F%F(F)#!\"*F'F(<#-F36#7$F(,*F7F(
F)#!$r\"F9*&F=F.FEF(F>F=FA" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 
"shm := matrix(2,2,[0, hm[1,2], -hm[2,1], 0]);" "6#>%$shmG-%'matrixG6%
\"\"#\"\"#7&\"\"!&%#hmG6$\"\"\"\"\"#,$&F-6$\"\"#\"\"\"!\"\"\"\"!" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$shmG-%'matrixG6#7$7$\"\"!,&\"#Q\"\"
\"%\"IG!#;7$,&!#QF-F.F/F*" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "
evshm := eigenvects(shm);" "6#>%&evshmG-%+eigenvectsG6#%$shmG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&evshmG6$7%,$*&%\"IG\"\"\"-%%sqrtG6#
\"#<\"\"\"\"#5F*<#-%'vectorG6#7$,&F(#!#>\"#&)*$F+F/#!\")F9F*7%,$F(!#5F
*<#-F36#7$,&F(#\"#>F9F:#\"\")F9F*" }}}{EXCHG {PARA 0 "> " 0 "" 
{XPPEDIT 19 1 "" "6#%#%?G" }}}}{MARK "25 0" 0 }{VIEWOPTS 1 1 0 1 1 
1803 }