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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 63 "Analog of diagonalizatio
n for a matrix with complex eigenvalues" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 37 "restart:\nA:=[[0,-2],[1,2]]:\nevalm(A);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$\"\"!!\"#7$\"\"\"\"\"#" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Q:=[[1,1],[-1,0]]:\nevalm(Q)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$\"\"\"F(7$!\"\"\"
\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "with(linalg):\nevalm
(inverse(Q)&*A&*Q);" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new defin
ition for norm" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition
 for trace" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$\"\"\"!
\"\"7$F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 40 "The solution (I al
lege) is Phi x0, where" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "A
:=[[a,-b],[b,a]]:\nevalm(A);\nPhi:=exp(a*t)*[[cos(b*t),-sin(b*t)],[sin
(b*t),cos(b*t)]]:\nevalm(Phi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'m
atrixG6#7$7$%\"aG,$%\"bG!\"\"7$F*F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#-%'matrixG6#7$7$*&-%$expG6#*&%\"aG\"\"\"%\"tGF.F.-%$cosG6#*&%\"bGF.F/
\"\"\"F.,$*&F)F5-%$sinGF2F.!\"\"7$F7F(" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 35 "Verification that ODE is satisfied:" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 74 "evalm(diff(Phi,t))-evalm(A&*Phi) = evalm( exp
and(diff(Phi,t) - A&*Phi) ) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%
'matrixG6#7$7$,&*(%\"aG\"\"\"-%$expG6#*&F,\"\"\"%\"tGF2F2-%$cosG6#*&%
\"bGF2F3F-F2F2*(F.F--%$sinGF6F2F8F-!\"\",&*(F,F-F.F-F:F-F<*(F.F-F4F-F8
F-F<7$,&F>F2F?F2F*F2-F&6#7$7$*&F.F-,&*&F,F-F4F-F2*&F:F-F8F-F<F2*&F.F-,
&*&F,F-F:F-F<*&F4F-F8F-F<F27$*&F.F-,&FMF2FLF2F2FFF<-F&6#7$7$\"\"!FUFT
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "Verification that initial con
dition is satisfied:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "eva
lm(eval(subs(t=0,Phi)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG
6#7$7$\"\"\"\"\"!7$F)F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 257 54 "For n
ext time: Diagonalization of the complexification" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 96 "evalm(A);\nevM:=transpose([[1,-I],[1,I]]):\n
evalm(evM);\nmap(expand,evalm( inverse(evM)&*A&*evM ));" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$%\"aG,$%\"bG!\"\"7$F*F(" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$\"\"\"F(7$,$%\"IG!\"\"
F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$,&%\"aG\"\"\"*&%
\"IGF*%\"bGF*F*\"\"!7$F.,&F)F*F+!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{MARK "10 0 0" 38 }{VIEWOPTS 1 1 0 1 1 1803 }