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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 79 "Solving a system with dis
tinct real eigenvalues by diagonalization: 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 transformation)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 210 "restart:\nb:=[[-1,0,0, 0],\n    [0, 2,0, 0],\n    [0, 0,-7,0],
\n    [0, 0,0, 4]];\nwith(linalg):\neigenvals(b);\nQ:=[[1,2,0,1],\n   \+
 [1,-1,0,3],\n    [1,1,2,1],\n    [0,0,-2,1]];\ninverse(Q);\nA:=evalm(
 Q&*b&*inverse(Q) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG7&7&!\"
\"\"\"!F(F(7&F(\"\"#F(F(7&F(F(!\"(F(7&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&!\"\"\"\"#!\"(\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%\"QG7&7&\"\"\"\"\"#\"\"!F'7&F'!\"\"F)\"\"$7&F'F'F(F'7&F)F)!\"#F'" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7&7&\"\"&\"\"$!\"(F*7&!\"
\"F,\"\"#F-7&F,#F,F-#F)F-\"\"\"7&!\"#F,F)F)" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"AG-%'matrixG6#7&7&!#<!#6\"#FF,7&!#F!#8\"#RF07&!\"\"
!\"#\"\"#\"\"*7&!#AF+\"#L\"#E" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 9 "evalm(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7&7&!#<!
#6\"#FF*7&!#F!#8\"#RF.7&!\"\"!\"#\"\"#\"\"*7&!#AF)\"#L\"#E" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "with(linalg):\neigenvals(A);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6&\"\"#\"\"%!\"(!\"\"" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 14 "eigenvects(A);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6&7%!\"(\"\"\"<#-%'vectorG6#7&\"\"!F+!\"\"F%7%\"\"%F%<#-F
(6#7&F%\"\"$F%F%7%F,F%<#-F(6#7&F%F%F%F+7%\"\"#F%<#-F(6#7&F:F,F%F+" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "evM:=transpose( [[],[],[],[]
] );" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 92 "evalm( inverse(evM)&*A&*evM );\nevalm( evM&*A&*invers
e(evM) ); # Note Hirsch and Smale wrong!" }}{PARA 11 "" 1 "" {TEXT -1 
0 "" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 182 "eajt:= [[exp(4*t),0,0,0],\n        [0,exp(-1*t),0,0]
,\n        [0,0,exp(2*t),0],\n        [0,0,0,exp(-7*t)]]:\narray(%);\n
x:= evalm(evM &* eajt &* inverse(evM) &* [[c1],[c2],[c3],[c4]]) ;" }
{TEXT -1 0 "" }}}}{MARK "4 0 0" 14 }{VIEWOPTS 1 1 0 1 1 1803 }