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urier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 }1 0 0 -1 -1 -1 0 0 0 0 0 0 
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0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple P
lot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 
-1 -1 -1 0 0 0 0 0 0 -1 0 }}
{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "First cook up example" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "restart:\nB:=[[3,-7,0,0,0,0
],\n    [7,3,0,0,0,0],\n    [1,0,3,-7,0,0],\n    [0,1,7,3,0,0],\n    [
0,0,0,0,5,0],\n    [0,0,0,0,1,5]]:\nevalm(B);\n    " }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#-%'matrixG6#7(7(\"\"$!\"(\"\"!F*F*F*7(\"\"(F(F*F*F*F*
7(\"\"\"F*F(F)F*F*7(F*F.F,F(F*F*7(F*F*F*F*\"\"&F*7(F*F*F*F*F.F1" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "Q:=[[1,2,3,4,1,2],\n    [2,
7,5,1,1,2],\n    [4,2,5,6,0,1],\n    [2,3,2,5,5,8],\n    [1,4,5,4,9,1]
,\n    [5,2,3,0,5,2]]:\nevalm(Q);\nwith(linalg):\nevalm(inverse(Q));\n
    " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7(7(\"\"\"\"\"#\"
\"$\"\"%F(F)7(F)\"\"(\"\"&F(F(F)7(F+F)F.\"\"'\"\"!F(7(F)F*F)F.F.\"\")7
(F(F+F.F+\"\"*F(7(F.F)F*F1F.F)" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning
, new definition for norm" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new
 definition for trace" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7
(7(#!%vH\"%oF#\"$0\"\"%%Q\"#\"$&oF-#\"$(GF-#\"#vF-#!$\"HF*7(#!%^XF*#\"
$8%F-#\"$\\)F-#\"$$[F-#\"$&HF-#!%$G\"F*7(#\"%z<\"$#p#!#v\"$Y$#!$T$FI#!
$0#FI#!$.\"FI#\"$b%FF7(#!%b9F-#\"#JFF#\"$n$FF#\"$x\"FF#\"$@\"FF#!$6&F-
7(#!$F'F*#!#VF-#\"#\\F-#\"#nF-#\"$n\"F-#!#RF*7(#\"%T9F-#!#fFF#!$>$FF#!
#pFF#!$T\"FF#\"$\"QF-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 262 65 "OK, her
e's the problem: solve the system x'=Ax with A as follows:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "A:=evalm(inverse(Q)&*B&*Q);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7(7(#\"&Lz%\"%oF#\"'
6O8F,#\"&\"oG\"$#p#\"%`*)F,#!%NmF1#!&z`#F,7(#\"&pM'F,#\"'Z`AF,#\"&&4ZF
1#\"&tM%F,#!%@#)F1#!&bH\"F,7(#!&(zCF1#!&JH)F1#!&Bt\"\"$t\"#!&pH\"F1#\"
%`NFL#\"%&*zF17(#\"&Xn\"\"%%Q\"#\"&*\\jFV#\"%*)oFL#\"%L%)FV#!%<CFL#!&$
G<FV7(#\"&h6\"F,#\"&JG$F,#\"%LoF1#\"%@sF,#\"%TNF1#\"%hRF,7(#!&>K\"FV#!
&XA'FV#!&6D\"\"$Y$#!$z#FV#\"%,KF_p#\"&xo\"FV" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 16 "eigenvectors(A);" }}{PARA 12 "" 1 "" {XPPMATH 
20 "6%7%\"\"&\"\"#<#-%'vectorG6#7(#\"$\"H\"%$G\"\"\"\"#!%?=F-#\"%A5F-#
\"#RF-#!$i(F-7%,&\"\"$F.%\"IG\"\"(F%<#-F(6#7(,&#\"&(RE\"%XMF.F:#\"&;f
\"FC,&#\"&\")p$FCF.F:#\"&3m\"FC,&#!&))3'FCF.F:#!&/c#FC,&#\"&U)HFCF.F:F
DF.,&#!%e:\"$l#F.F:#!%%Q\"FV7%,&F9F.F:!\"(F%<#-F(6#7(,&FAF.F:#!&;f\"FC
,&FGF.F:#!&3m\"FC,&FLF.F:#\"&/c#FC,&FQF.F:F[oF.,&FTF.F:#\"%%Q\"FV" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "id6:=array(identity, 1..6,1.
.6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$id6G-%&arrayG6&%)identityG;
\"\"\"\"\"'F)7\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 171 "basis1
:= kernel( (A-(3+7*I)*id6) ^2 );\nbasis11:=basis1[1];\nim11:=map(Im,ba
sis11);\nre11:=map(Re,basis11);\nbasis12:=basis1[2];\nim12:=map(Im,bas
is12);\nre12:=map(Re,basis12);\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%
'basis1G<$-%'vectorG6#7(,&#!&,y\"\"&Yd$\"\"\"%\"IG#\"%Xq\"&ty\",&#\"'$
R$>\"'#o2'F.F/#!&IK)\"'TQI\"\"!,&#!'!z9\"F9F.F/#\"'`yCF9,&#!&8y\"F6F.F
/#!&tk)F9F.-F'6#7(,&#!%LxF-F.F/#!%>bF-,&#!&AQ\"F2F.F/#\"$&))F2F.,&#!%F
\"*F-F.F/#!&:7\"F-,&#!%dEF-F.F/#\"%6[F-F:" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%(basis11G-%'vectorG6#7(,&#!&,y\"\"&Yd$\"\"\"%\"IG#\"%
Xq\"&ty\",&#\"'$R$>\"'#o2'F-F.#!&IK)\"'TQI\"\"!,&#!'!z9\"F8F-F.#\"'`yC
F8,&#!&8y\"F5F-F.#!&tk)F8F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%im11
G-%'vectorG6#7(#\"%Xq\"&ty\"#!&IK)\"'TQI\"\"!#\"'`yCF.#!&tk)F.F/" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%re11G-%'vectorG6#7(#!&,y\"\"&Yd$#\"
'$R$>\"'#o2'\"\"!#!'!z9\"\"'TQI#!&8y\"F.\"\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%(basis12G-%'vectorG6#7(,&#!%Lx\"&Yd$\"\"\"%\"IG#!%>bF
,,&#!&AQ\"\"&ty\"F-F.#\"$&))F4F-,&#!%F\"*F,F-F.#!&:7\"F,,&#!%dEF,F-F.#
\"%6[F,\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%im12G-%'vectorG6#7(
#!%>b\"&Yd$#\"$&))\"&ty\"\"\"!#!&:7\"F+#\"%6[F+F/" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%re12G-%'vectorG6#7(#!%Lx\"&Yd$#!&AQ\"\"&ty\"\"\"\"#!
%F\"*F+#!%dEF+\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "id6
:=array(identity, 1..6,1..6):\nbasis2:= kernel( (A-(5)*id6) ^2 );\nbas
is21:=basis2[1];\nbasis22:=basis2[2];\n" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%'basis2G<$-%'vectorG6#7(\"\"\"#!#8\"\"*#\"#GF-\"\"!#\"$L#F-#!#
]\"\"$-F'6#7(F0#\"\"&F5#!\")F5F*#!#AF5\"\"%" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%(basis21G-%'vectorG6#7(\"\"\"#!#8\"\"*#\"#GF,\"\"!#\"
$L#F,#!#]\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(basis22G-%'vector
G6#7(\"\"!#\"\"&\"\"$#!\")F,\"\"\"#!#AF,\"\"%" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 6 "?array" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "T
his horrible construction is the only way I could figure out to automa
te the building of the similarity matrix, P:" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 289 "P:=array(1..6,1..6);\nfor i from 1 to 6 do P[i,
1]:=im11[i]: od:\nfor i from 1 to 6 do P[i,2]:=re11[i]: od:\nfor i fro
m 1 to 6 do P[i,3]:=im12[i]: od:\nfor i from 1 to 6 do P[i,4]:=re12[i]
: od:\nfor i from 1 to 6 do P[i,5]:=basis21[i]: od:\nfor i from 1 to 6
 do P[i,6]:=basis22[i]: od:\nevalm(P);\n" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"PG-%&arrayG6%;\"\"\"\"\"'F(7\"" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#-%'matrixG6#7(7(#\"%Xq\"&ty\"#!&,y\"\"&Yd$#!%>bF-#!%LxF-\"\"\"\"
\"!7(#!&IK)\"'TQI#\"'$R$>\"'#o2'#\"$&))F*#!&AQ\"F*#!#8\"\"*#\"\"&\"\"$
7(F3F3F3F2#\"#GFA#!\")FD7(#\"'`yCF7#!'!z9\"F7#!&:7\"F-#!%F\"*F-F3F27(#
!&tk)F7#!&8y\"F:#\"%6[F-#!%dEF-#\"$L#FA#!#AFD7(F3F2F3F3#!#]FD\"\"%" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 183 "S1:=[[ 3,-7,0,0,0,0],\n   \+
  [ 7, 3,0,0,0,0],\n     [ 0, 0,3,-7,0,0],\n     [ 0, 0,7, 3,0,0],\n  \+
   [ 0, 0,0, 0,5,0],\n     [ 0, 0,0, 0,0,5]]: evalm(S1);\nN:=evalm( A \+
-  P&*S1&*inverse(P) );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7(7(\"\"$!\"(\"\"!F*F*F*7(
\"\"(F(F*F*F*F*7(F*F*F(F)F*F*7(F*F*F,F(F*F*7(F*F*F*F*\"\"&F*7(F*F*F*F*
F*F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG-%'matrixG6#7(7(#\"%FA\"
%oF#\"%(z#\"%%Q\"#\"%DbF,#\"%XCF/#!$v'F,#\"%(f$F,7(#\"%ZBF,#\"%8DF/#\"
%4NF,#\"%88F/#!%$)))F,#\"%XSF,7(#!%Z5\"$#p#!%27\"$Y$#!%@=FH#!$f'FK#\"%
.IFH#!%H<FH7(#\"$J*F/#\"$^*FH#\"%<9F/#\"$B'FH#!%6NF/#\"%l;F/7(#\"$F$F,
#\"$*[F/#\"$p(F,#\"$&=F/#!$>\"F,#\"$D%F,7(#!$L&F/#!$f$FH#!$*pF/#!$$eFH
#\"%`EF/#!%r6F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "evalm(N^
2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7(7(\"\"!F(F(F(F(F(
F'F'F'F'F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "I use \"exponentia
l\" here to save time, but realize that this exponential can be writte
n down\nwithout any calculation:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "expS1t:=exponential(S1*t);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%'expS1tG-%'matrixG6#7(7(*&-%$expG6#,$%\"tG\"\"$\"\"\"
-%$cosG6#,$F/\"\"(F1,$*&F+\"\"\"-%$sinGF4F1!\"\"\"\"!F=F=F=7(F8F*F=F=F
=F=7(F=F=F*F7F=F=7(F=F=F8F*F=F=7(F=F=F=F=-F,6#,$F/\"\"&F=7(F=F=F=F=F=F
B" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "expNt:=evalm(id6 + N*t
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&expNtG-%'matrixG6#7(7(,&\"\"
\"F+%\"tG#\"%FA\"%oF,$F,#\"%(z#\"%%Q\",$F,#\"%DbF/,$F,#\"%XCF3,$F,#!$v
'F/,$F,#\"%(f$F/7(,$F,#\"%ZBF/,&F+F+F,#\"%8DF3,$F,#\"%4NF/,$F,#\"%88F3
,$F,#!%$)))F/,$F,#\"%XSF/7(,$F,#!%Z5\"$#p,$F,#!%27\"$Y$,&F+F+F,#!%@=FW
,$F,#!$f'Fen,$F,#\"%.IFW,$F,#!%H<FW7(,$F,#\"$J*F3,$F,#\"$^*FW,$F,#\"%<
9F3,&F+F+F,#\"$B'FW,$F,#!%6NF3,$F,#\"%l;F37(,$F,#\"$F$F/,$F,#\"$*[F3,$
F,#\"$p(F/,$F,#\"$&=F3,&F+F+F,#!$>\"F/,$F,#\"$D%F/7(,$F,#!$L&F3,$F,#!$
f$FW,$F,#!$*pF3,$F,#!$$eFW,$F,#\"%`EF3,&F+F+F,#!%r6F3" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "expAt:= simplify(evalm(P&*expS1t&*i
nverse(P)&*expNt));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&expAtG-%'mat
rixG6#7(7(,.*&-%$expG6#,$%\"tG\"\"$\"\"\"-%$cosG6#,$F0\"\"(F2#\"%tS\"%
oF*(F0F2F,\"\"\"F3F<#\"%f7\"%%Q\"*&F,F<-%$sinGF5F2#\"%H9\"$#p*(F0F<F,F
<FAF<#!%$3\"F?-F-6#,$F0\"\"&#!%08F:*&F0F<FIF2#!$\"HF:,.F;#\"%zLF?FF#!%
@UF?FO#FQFEF+#!\"*F?F@#\"&$G=F:FI#\"\"*F?,.F;#\"%X<FEFF#!$T'\"$Y$FO#!%
b9F:F+#\"$B\"F:F@#\"&Nc\"F:FI#!$B\"F:,.F;#\"%FIF?FF#\"$j%F?FOFWF+#!#vF
]oF@#\"$4%F:FI#\"#vF]o,.F;#\"$V#F]oFF#!$*>FEFO#!%>EF:F+#\"$0\"F:F@#!%l
OF:FI#!$0\"F:,.F;#Fcp\"$t\"FF#FepF]oFOFPF+#\"#FF`qF@#!$^#F`qFI#!#FF`q7
(,.F+#\"%DeF:F;#\"%:=F?F@#\"%*f#FEFF#!%:7F?FI#!%DeF:FO#!%$G\"F:,.F;#\"
%z]F?FF#!%x\\F?FO#FerFEF+#\"%([\"F?F@#\"&Z.$F:FI#!$.\"F?,.F;#\"%\"[#FE
FF#!$*pF]oFO#!%:kF:F+#\"$**)F:F@#\"&nm#F:FI#!$**)F:,.F;#\"%zQF?FF#\"%$
3\"F?FOF[sF+#!$&HF]oF@#\"%h^F:FI#\"$&HF]o,.F;#\"$L$F]oFF#!$$=FEFO#!&Z:
\"F:F+#\"%06F:F@#!%8JF:FI#!%06F:,.F;#F\\uF`qFF#F^uF]oFOFdrF+#\"$Z#F]oF
@#!$L#F]oFI#!$Z#F]o7(,.F@#!$'**F`qFF#\"$x%F]oF+#!%p?FEF;#!$^(F]oFI#\"%
p?FEFO#\"$b%FE,.FF#\"%x>F]oF;#!%<@F]oFO#F]wF`qF@#!&F:\"FEF+#!#VF]oFI#
\"#VF]o,.FF#\"$X&F`qF;#!%C5F`qFO#\"%vAFEF@#!&J+\"FEF+#\"$d$FEFI#\"$N$F
E,.FF#!$z%F]oF;#!%p:F]oFOFcwF@#!%H9FEF+#\"$1#F`qFI#!$1#F`q,.FF#\"#oF`q
F;#!$t#F`qFO#\"%&4%FEF@#\"%\"[\"FEF+#!$@%FEFI#\"$@%FE,.FF#\"$O\"F`qF;#
!$Y&F`qFOF\\wF@#\"$(HF`qF+#!$w\"F`qFI#\"$w\"F`q7(,.F+#\"%8BF?F;#\"$@(F
EF@#\"$l$F`qFF#!$d&FEFI#!%8BF?FO#!$6&F?,.F;#\"%t>FEFF#!%:AFEFO#Fg[lF]o
F+#FcqFEF@#\"%:))F?FI#FgqFE,.F;#\"$$**F]oFF#!$E$F`qFO#!%bDF?F+#\"$B$F?
F@#\"%jxF?FI#!$B$F?,.F;#\"%X;FEFF#\"$T$FEFOF]\\lF+#\"#_F`qF@#\"$d\"F?F
I#\"$@\"F`q,.F;F`zFF#!#&*F]oFO#!%*f%F?F+#\"$x$F?F@#!%`@F?FI#!$x$F?,.F;
#\"$s#F`qFF#F\\^lF`qFOFf[lF+#\"$&>F]oF@#!$@'F]oFI#!$&>F]o7(,.F+#!$R\"F
:F;#\"$$=F?F@#F`^lFEFF#!#JF?FI#\"$R\"F:FO#!#RF:,.F;#\"$n&F?FF#!$4#F?FO
#F[`lFEF+#!$H'F?F@#\"%\">%F:FI#\"$H'F?,.F;#\"$T#FEFF#!#6F]oFO#F^_lF:F+
#!%`:F:F@#\"%^LF:FI#\"%`:F:,.F;#\"$j#F?FF#\"$>#F?FOFa`lF+#!$n\"F]oF@#
\"$(fF:FI#\"$n\"F]o,.F;#\"#HF]oFF#FgnFEFO#!$^$F:F+#FgwF:F@#\"#^F:FI#\"
%6GF:,.F;#FablF`qFF#FgnF]oFOFj_lF+#!#;F`qF@FbqFI#\"#;F`q7(,.F@#!$p&F]o
FF#\"$p&FEF+#!%B;F?F;#!$d%FEFI#\"%B;F?FO#\"$\"QF?,.FF#\"%&4#FEF;#!%@6F
EFO#F^dlF]oF@#!%&o)F?F+#Fd_lFEFIF]u,.FF#\"$Z$F`qF;#!$^'F]oFO#\"%0>F?F@
#!%tpF?F+#\"$n#F?FI#!$n#F?,.FF#FiwFEF;#!%X8FEFOFddlF@#\"$\\%F?F+#\"$T
\"F`qFI#!$T\"F`q,.FF#\"$D\"F]oF;#!#(*F`qFO#\"%HMF?F@#\"%J;F?F+#\"$L'F?
FI#!$L'F?,.FF#FaflF`qF;#!$%>F`qFOF]dlF@#\"$J#F`qF+#\"$8\"F`qFI#\"#gF`q
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "Should_Be_Zero := simpl
ify( evalm( map(diff,expAt,t) - A &* expAt ));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%/Should_Be_ZeroG-%'matrixG6#7(7(\"\"!F*F*F*F*F*F)F)F)
F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "Just for fun, choose an i
nitial condition and plot the components of the solution:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 141 "x0:=[[1],[1],[2],[-1],[0],[3]]: ev
alm(x0);\nxt:=evalm(expAt&*x0);\nplot( \{xt[1,1],xt[2,1],xt[3,1],xt[4,
1],xt[5,1],xt[6,1]\}, t=0..1 ,-200..200);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'matrixG6#7(7#\"\"\"F'7#\"\"#7#!\"\"7#\"\"!7#\"\"$" }
}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#xtG-%'matrixG6#7(7#,.*&%\"tG\"\"\"
-%$expG6#,$F,\"\"&F-#!%P?\"%%Q\"*(F,\"\"\"-F/6#,$F,\"\"$F--%$cosG6#,$F
,\"\"(F-#\"&BW\"F5F.#!%HM\"%oF*&F8F7F<F7#\"%(>'FE*&F8F7-%$sinGF>F-#\"&
.2\"\"$#p*(F,F7F8F7FJF7#!&$G8F57#,.F+#!%\")*)F5F6#\"&J4#F5F.#!&<h\"FEF
F#\"&&))=FEFI#\"&J3#FNFO#!&j]\"F57#,.F+#\"%&=$\"$Y$F6#!%r')F^oF.#\"%hd
FNFF#!%xVFNFI#!%Xw\"$t\"FO#\"%HfF^o7#,.F+#!%xNFNF6#\"%&G)FNF.#!%@jF5FF
#\"%P\\F5FI#\"%8\\F^oFO#!%hoFN7#,.F+#!$t#F5F6#\"%Z@F5F.#\"%NRFEFF#!%NR
FEFI#\"%vKFNFO#!$R%F57#,.F+#\"%nEFNF6#!%l^FNF.#\"%\"H$F5FF#\"$h)F5FI#!
%`\\F^oFO#\"%(*oFN" }}{PARA 13 "" 1 "" {INLPLOT "6*-%'CURVESG6$7W7$\"
\"!F(7$$\"1nmm;arz@!#<$\"1%yEsR)RH))!#;7$$\"1LL$e9ui2%F,$\"1XPc\\zYh<!
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