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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 63 "Solving a system with rea
l and complex eigenvalues: an example." }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 139 "First cook up the example matrix, A (by taking one that'
s already diagonalized, and disguising it by applying a similarity tra
nsformation)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 262 "restart:
\nb:=[[-1,-4,0, 0,0],\n    [4,-1,0, 0,0],\n    [0, 0,-7,3,0],\n    [0,
 0,-3,-7,0],\n    [0,0,0,0,2]];\nwith(linalg):\neigenvals(b);\nQ:=[[1,
2,0,1,1],\n    [1,-1,0,3,0],\n    [1,1,2,1,0],\n    [0,0,-2,1,1],\n   \+
 [1,0,0,1,2]];\ninverse(Q);\nA:=evalm( Q&*b&*inverse(Q) );" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"bG7'7'!\"\"!\"%\"\"!F)F)7'\"\"%F'F)F)F)7
'F)F)!\"(\"\"$F)7'F)F)!\"$F-F)7'F)F)F)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'\"\"#,&!\"\"\"\"\"%\"IG\"\"%,&F%F&F'!\"%,&!\"(F&F'\"\"$
,&F,F&F'!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"QG7'7'\"\"\"\"\"#
\"\"!F'F'7'F'!\"\"F)\"\"$F)7'F'F'F(F'F)7'F)F)!\"#F'F'7'F'F)F)F'F(" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7'7'\"\"$#\"\"&F(#!#8F(F+#
\"\"#F(7'\"\"!#!\"\"F(F-F-F17'F2#F2F.#F(F.\"\"\"F07'F2F1F)F)F17'F2#!\"
#F(#\"\"%F(F;#F6F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG
6#7'7'\"#H#\"#(*\"\"'#!$(GF-#!$R\"\"\"$#F*F27'\"#:#\"#DF-#!$t\"F-#!#tF
2#\"#9F27'\"#F#\"#J\"\"##!#()FB!#N\"\"%7'\"\"!#!\"&FBFI!\")\"\"&7'F2#F
LF-#!#bF-#!#BF2#\"#8F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "ev
alm(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7'7'\"#H#\"#(*
\"\"'#!$(GF+#!$R\"\"\"$#F(F07'\"#:#\"#DF+#!$t\"F+#!#tF0#\"#9F07'\"#F#
\"#J\"\"##!#()F@!#N\"\"%7'\"\"!#!\"&F@FG!\")\"\"&7'F0#FJF+#!#bF+#!#BF0
#\"#8F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "with(linalg):\ne
igenvals(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'\"\"#,&!\"\"\"\"\"%\"I
G\"\"%,&F%F&F'!\"%,&!\"(F&F'\"\"$,&F,F&F'!\"$" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 14 "eigenvects(A);" }}{PARA 12 "" 1 "" {XPPMATH 20 "
6'7%,&!\"(\"\"\"%\"IG\"\"$F&<#-%'vectorG6#7',&#F&\"\"&F&F'#!\"#F0,&#F(
F0F&F'#!\"'F0,&#!\"$F0F&F'#!\"%F0F&F.7%,&F%F&F'F9F&<#-F+6#7',&F/F&F'#
\"\"#F0,&F4F&F'#\"\"'F0,&F8F&F'#\"\"%F0F&FB7%FDF&<#-F+6#7'F&\"\"!FPF&F
D7%,&!\"\"F&F'FJF&<#-F+6#7',&F&F&F'F2,&F&F&F'F&,&F&F&F'FSFPF&7%,&FSF&F
'F;F&<#-F+6#7',&F&F&F'FDFZFYFPF&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 258 
40 "Diagonalization of the complexification:" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 233 "evM:=transpose( [[1-2*I,3-6*I,-3-4*I,5,1-2*I],
\n                 [1+2*I,3+6*I,-3+4*I,5,1+2*I],\n                 [1,
0,0,1,2],\n                 [1-2*I,1+I,1-I,0,1],\n                 [1+
2*I,1-I,1+I,0,1]] );\nevalm( inverse(evM)&*A&*evM );" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%$evMG-%'matrixG6#7'7',&\"\"\"F+%\"IG!\"#,&F+F+F,\"
\"#F+F*F.7',&\"\"$F+F,!\"',&F2F+F,\"\"'\"\"!,&F+F+F,F+,&F+F+F,!\"\"7',
&!\"$F+F,!\"%,&F<F+F,\"\"%F6F8F77'\"\"&FAF+F6F67'F*F.F/F+F+" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7'7',&!\"(\"\"\"%\"IG\"\"$\"\"!F
-F-F-7'F-,&F)F*F+!\"$F-F-F-7'F-F-\"\"#F-F-7'F-F-F-,&!\"\"F*F+\"\"%F-7'
F-F-F-F-,&F5F*F+!\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 259 27 "Real blo
ck-diagonalization:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 108 "Pick off \+
the real eigenvectors and the real and imaginary parts of one of each \+
pair of nonreal eigenvectors:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 87 "evM:=transpose( [[-2,-6,-4,0,-2],[1,3,-3,5,1],[-2,1,-1,0,0],[1,1
,1,0,1],[1,0,0,1,2] ]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$evMG-%'m
atrixG6#7'7'!\"#\"\"\"F*F+F+7'!\"'\"\"$F+F+\"\"!7'!\"%!\"$!\"\"F+F/7'F
/\"\"&F/F/F+7'F*F+F/F+\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
30 "evalm( inverse(evM)&*A&*evM );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
-%'matrixG6#7'7'!\"(!\"$\"\"!F*F*7'\"\"$F(F*F*F*7'F*F*!\"\"!\"%F*7'F*F
*\"\"%F.F*7'F*F*F*F*\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 
"" }}}}{MARK "12 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }