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0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "
" {TEXT 257 33 "Sistemas de Equa\347\365es de Diferen\347a" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "Um problema:" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 105 "Queremos
 contar quantos caminhos (fechados) de tamanho n come\347am e terminam
 no v\351rtice 1 de um tri\342ngulo." }}}{EXCHG {PARA 0 "> " 0 "" 
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{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "Por simetria, vemos que come\347ar
 e terminar em 1 ou em 2 ou em 3 d\341 no mesmo." }}{PARA 0 "" 0 "" 
{TEXT -1 36 "Chamemos essa quantidade, ent\343o, de " }{XPPEDIT 18 0 "
F[n]" "6#&%\"FG6#%\"nG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 17 
"De modo an\341logo, " }{XPPEDIT 18 0 "A[n]" "6#&%\"AG6#%\"nG" }{TEXT 
-1 73 " ser\341 o n\372mero de caminhos (abertos) indo de um v\351rtic
e a outro distinto." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 176 "Queremos ver se conseguimos escrever uma equa\347\343o
 para cada vari\341vel que relacione os caminhos de tamanho n com os d
e tamanho n-1. A id\351ia \351 dividir os caminhos de tamanho n em " }
{TEXT 258 17 "classes distintas" }{TEXT -1 33 " e contar quantos h\341
 em cada uma." }}{PARA 0 "" 0 "" {TEXT -1 82 "Por exemplo, vamos lista
r os caminhos fechados com in\355cio e fim em 1 de tamanho 4:" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "1, 2, 3, \+
2, 1" }}{PARA 0 "" 0 "" {TEXT -1 13 "1, 3, 2, 3, 1" }}{PARA 0 "" 0 "" 
{TEXT -1 13 "1, 2, 1, 2, 1" }}{PARA 0 "" 0 "" {TEXT -1 13 "1, 3, 1, 3,
 1" }}{PARA 0 "" 0 "" {TEXT -1 13 "1, 3, 1, 2, 1" }}{PARA 0 "" 0 "" 
{TEXT -1 13 "1, 2, 1, 3, 1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 109 "Vamos olhar agora para os primeiros n-1(=3 no ca
so) passos. O importante \351 ver onde terminamos ap\363s 3 passos." }
}{PARA 0 "" 0 "" {TEXT -1 49 "Vou marcar de cores distintas destinos d
istintos:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
9 "1, 2, 3, " }{TEXT 259 1 "2" }{TEXT -1 3 ", 1" }}{PARA 0 "" 0 "" 
{TEXT -1 9 "1, 3, 2, " }{TEXT 262 1 "3" }{TEXT -1 3 ", 1" }}{PARA 0 "
" 0 "" {TEXT -1 9 "1, 2, 1, " }{TEXT 260 1 "2" }{TEXT -1 3 ", 1" }}
{PARA 0 "" 0 "" {TEXT -1 9 "1, 3, 1, " }{TEXT 263 1 "3" }{TEXT -1 3 ",
 1" }}{PARA 0 "" 0 "" {TEXT -1 9 "1, 3, 1, " }{TEXT 261 1 "2" }{TEXT 
-1 3 ", 1" }}{PARA 0 "" 0 "" {TEXT -1 9 "1, 2, 1, " }{TEXT 264 1 "3" }
{TEXT -1 3 ", 1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 97 "Note que n\343o existe nenhum caminho de 4 passos de 1 a \+
1 que ap\363s 3 passos esteja em 1 (por que?)." }}{PARA 0 "" 0 "" 
{TEXT -1 89 "H\341 tantos caminhos vermelhos quanto caminhos abertos d
e 3 passos indo de 1 a 2, ou seja, " }{XPPEDIT 18 0 "A[3]" "6#&%\"AG6#
\"\"$" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 84 "De modo an\341lo
go, h\341 tantos azuis quantos caminhos abertos de 1 a 3, ou seja, tam
b\351m " }{XPPEDIT 18 0 "A[3]" "6#&%\"AG6#\"\"$" }{TEXT -1 2 ". " }}
{PARA 0 "" 0 "" {TEXT -1 45 "Como acima todos os caminhos est\343o lis
tados, " }{XPPEDIT 18 0 "F[4] = A[3]+A[3];" "6#/&%\"FG6#\"\"%,&&%\"AG6
#\"\"$\"\"\"&F*6#F,F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 59 "
O argumento pode ser feito de forma geral e conclu\355mos que " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "F[n]=
2*A[n]" "6#/&%\"FG6#%\"nG*&\"\"#\"\"\"&%\"AG6#F'F*" }{TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 25 "Se quisermos saber agora " }{XPPEDIT 18 
0 "A[4]" "6#&%\"AG6#\"\"%" }{TEXT -1 42 ", digamos de 1 a 2, fa\347amo
s a mesma coisa:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 9 "1, 2, 3, " }{TEXT 265 1 "1" }{TEXT -1 3 ", 2" }}{PARA 0 "
" 0 "" {TEXT -1 8 "1, 3, 2," }{TEXT 266 1 " " }{TEXT 270 1 "1" }{TEXT 
-1 3 ", 2" }}{PARA 0 "" 0 "" {TEXT -1 9 "1, 2, 1, " }{TEXT 268 1 "3" }
{TEXT -1 3 ", 2" }}{PARA 0 "" 0 "" {TEXT -1 9 "1, 3, 2, " }{TEXT 267 
1 "1" }{TEXT -1 3 ", 2" }}{PARA 0 "" 0 "" {TEXT -1 9 "1, 3, 1, " }
{TEXT 269 1 "3" }{TEXT -1 3 ", 2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 99 "Agora o que n\343o consta s\343o caminhos
 que ap\363s 3 passos estejam em 2, nosso destino final. Novamente " }
{XPPEDIT 18 0 "A[4]" "6#&%\"AG6#\"\"%" }{TEXT -1 43 " ser\341 a soma d
e caminhos vermelhos e azuis." }}{PARA 0 "" 0 "" {TEXT -1 80 "Os verme
lhos s\343o em mesmo n\372mero que os fechados de 1 a 1 de 3 passos, o
u seja, " }{XPPEDIT 18 0 "F[3]" "6#&%\"FG6#\"\"$" }{TEXT -1 78 " e os \+
azuis s\343o em mesmo n\372mero que os abertos de 1 a 3 em 3 passos, o
u seja, " }{XPPEDIT 18 0 "A[3]" "6#&%\"AG6#\"\"$" }{TEXT -1 2 ": " }
{XPPEDIT 18 0 "A[4]=A[3]+F[3]" "6#/&%\"AG6#\"\"%,&&F%6#\"\"$\"\"\"&%\"
FG6#F+F," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 28 "Novamente, va
le que em geral" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" 
{XPPEDIT 18 0 "A[n]=A[n-1]+F[n-1]" "6#/&%\"AG6#%\"nG,&&F%6#,&F'\"\"\"F
,!\"\"F,&%\"FG6#,&F'F,F,F-F," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "Obtemos assim o seguinte sistem
a de ED's:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{XPPEDIT 18 0 "A[n]=A[n-1]+F[n-1]" "6#/&%\"AG6#%\"nG,&&F%6#,&F'\"\"\"F
,!\"\"F,&%\"FG6#,&F'F,F,F-F," }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "F[n]=2*
A[n]" "6#/&%\"FG6#%\"nG*&\"\"#\"\"\"&%\"AG6#F'F*" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "com " }{XPPEDIT 18 0 "A[0]
=0" "6#/&%\"AG6#\"\"!F'" }{TEXT -1 3 " e " }{XPPEDIT 18 0 "F[0]=1" "6#
/&%\"FG6#\"\"!\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 52 "Se quisermos, digamos, F[11], podemos f
azer um loop:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "# inicializ\nA[0]:=0;\nF[0
]:=1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"AG6#\"\"!F'" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>&%\"FG6#\"\"!\"\"\"" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 81 "# e faz loop\nfor n from 1 to 11 do\n A[n]:=A[n-1]
+F[n-1];\n F[n]:=2*A[n-1];\nend do;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>&%\"AG6#\"\"\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"FG6#\"\"\"
\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"AG6#\"\"#\"\"\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"FG6#\"\"#F'" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%\"AG6#\"\"$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%
\"FG6#\"\"$\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"AG6#\"\"%\"\"
&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"FG6#\"\"%\"\"'" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>&%\"AG6#\"\"&\"#6" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>&%\"FG6#\"\"&\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"AG6#
\"\"'\"#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"FG6#\"\"'\"#A" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"AG6#\"\"(\"#V" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>&%\"FG6#\"\"(\"#U" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>&%\"AG6#\"\")\"#&)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"FG6#\"\")
\"#')" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"AG6#\"\"*\"$r\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"FG6#\"\"*\"$q\"" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>&%\"AG6#\"#5\"$T$" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>&%\"FG6#\"#5\"$U$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"AG6#\"#6
\"$$o" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"FG6#\"#6\"$#o" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "Para equa\347\365es escalares, t
\355nhamos rsolve. Vamos ver se d\341 certo aqui tamb\351m:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "unassign('n');\nrsolve(\{a(n
)=a(n-1)+f(n-1),f(n)=2*a(n-1),a(0)=0, f(0)=1\},\{a(n), f(n)\});" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/-%\"aG6#%\"nG,&*&\"\"$!\"\")F,F(\"
\"\"F,*&F+F,)\"\"#F(F.F./-%\"fGF',&*(F1F.F+F,F-F.F.*&F+F,F0F.F." }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "Antes de obtermos essa f\363rmula,
 um pouco de estrutura." }}{PARA 0 "" 0 "" {TEXT -1 83 "Podemos escrev
er o sistema de duas equa\347\365es como uma \372nica igualdade entre \+
vetores:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" 
{XPPEDIT 18 0 "RTABLE(86601808,MATRIX([[A[n]], [F[n]]]),Vector[column]
) = RTABLE(86627784,MATRIX([[A[n-1]+F[n-1]], [2*A[n-1]]]),Vector[colum
n]);" "6#/-%'RTABLEG6%\")3=g')-%'MATRIXG6#7$7#&%\"AG6#%\"nG7#&%\"FGF/&
%'VectorG6#%'columnG-F%6%\")%yFm)-F)6#7$7#,&&F.6#,&F0\"\"\"FC!\"\"FC&F
3FAFC7#,$*&\"\"#FCF@FCFCF4" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "RTABLE(8
6077512,MATRIX([[1, 1], [2, 0]]),Matrix);" "6#-%'RTABLEG6%\")7v2')-%'M
ATRIXG6#7$7$\"\"\"F,7$\"\"#\"\"!%'MatrixG" }{TEXT -1 1 " " }{XPPEDIT 
18 0 "RTABLE(86596232,MATRIX([[A[n-1]], [F[n-1]]]),Vector[column]);" "
6#-%'RTABLEG6%\")Kif')-%'MATRIXG6#7$7#&%\"AG6#,&%\"nG\"\"\"F1!\"\"7#&%
\"FGF.&%'VectorG6#%'columnG" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "de modo que, definindo" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {XPPEDIT 18 0 "v[n] \+
= RTABLE(86849556,MATRIX([[A[n]], [F[n]]]),Vector[column]);" "6#/&%\"v
G6#%\"nG-%'RTABLEG6%\")c&\\o)-%'MATRIXG6#7$7#&%\"AGF&7#&%\"FGF&&%'Vect
orG6#%'columnG" }{TEXT -1 5 ",  M=" }{XPPEDIT 18 0 "RTABLE(86077512,MA
TRIX([[1, 1], [2, 0]]),Matrix)" "6#-%'RTABLEG6%\")7v2')-%'MATRIXG6#7$7
$\"\"\"F,7$\"\"#\"\"!%'MatrixG" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" 
{TEXT -1 47 "vemos que o sistema pode ser escrito ent\343o como" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "v[n]=
M*v[n-1]" "6#/&%\"vG6#%\"nG*&%\"MG\"\"\"&F%6#,&F'F*F*!\"\"F*" }{TEXT 
-1 5 ",    " }{XPPEDIT 18 0 "v[0] = RTABLE(86379832,MATRIX([[0], [1]])
,Matrix);" "6#/&%\"vG6#\"\"!-%'RTABLEG6%\")K)zj)-%'MATRIXG6#7$7#F'7#\"
\"\"%'MatrixG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 37 "cuja solu\347\343o \351, como no caso esc
alar," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {XPPEDIT 18 
0 "v[n]=M^n*v[0]" "6#/&%\"vG6#%\"nG*&)%\"MGF'\"\"\"&F%6#\"\"!F+" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 46 "#pacote de \341lgebra linear\nwith(LinearAlgebra
);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7er%#&xG%$AddG%(AdjointG%3Backwa
rdSubstituteG%+BandMatrixG%&BasisG%-BezoutMatrixG%/BidiagonalFormG%-Bi
linearFormG%5CharacteristicMatrixG%9CharacteristicPolynomialG%'ColumnG
%0ColumnDimensionG%0ColumnOperationG%,ColumnSpaceG%0CompanionMatrixG%0
ConditionNumberG%/ConstantMatrixG%/ConstantVectorG%%CopyG%2CreatePermu
tationG%-CrossProductG%-DeleteColumnG%*DeleteRowG%,DeterminantG%)Diago
nalG%/DiagonalMatrixG%*DimensionG%+DimensionsG%+DotProductG%6EigenCond
itionNumbersG%,EigenvaluesG%-EigenvectorsG%&EqualG%2ForwardSubstituteG
%.FrobeniusFormG%4GaussianEliminationG%2GenerateEquationsG%/GenerateMa
trixG%(GenericG%2GetResultDataTypeG%/GetResultShapeG%5GivensRotationMa
trixG%,GramSchmidtG%-HankelMatrixG%,HermiteFormG%3HermitianTransposeG%
/HessenbergFormG%.HilbertMatrixG%2HouseholderMatrixG%/IdentityMatrixG%
2IntersectionBasisG%+IsDefiniteG%-IsOrthogonalG%*IsSimilarG%*IsUnitary
G%2JordanBlockMatrixG%+JordanFormG%1KroneckerProductG%(LA_MainG%0LUDec
ompositionG%-LeastSquaresG%,LinearSolveG%$MapG%%Map2G%*MatrixAddG%2Mat
rixExponentialG%/MatrixFunctionG%.MatrixInverseG%5MatrixMatrixMultiply
G%+MatrixNormG%,MatrixPowerG%5MatrixScalarMultiplyG%5MatrixVectorMulti
plyG%2MinimalPolynomialG%&MinorG%(ModularG%)MultiplyG%,NoUserValueG%%N
ormG%*NormalizeG%*NullSpaceG%3OuterProductMatrixG%*PermanentG%&PivotG%
*PopovFormG%0QRDecompositionG%-RandomMatrixG%-RandomVectorG%%RankG%6Ra
tionalCanonicalFormG%6ReducedRowEchelonFormG%$RowG%-RowDimensionG%-Row
OperationG%)RowSpaceG%-ScalarMatrixG%/ScalarMultiplyG%-ScalarVectorG%*
SchurFormG%/SingularValuesG%*SmithFormG%8StronglyConnectedBlocksG%*Sub
MatrixG%*SubVectorG%)SumBasisG%0SylvesterMatrixG%/ToeplitzMatrixG%&Tra
ceG%*TransposeG%0TridiagonalFormG%+UnitVectorG%2VandermondeMatrixG%*Ve
ctorAddG%,VectorAngleG%5VectorMatrixMultiplyG%+VectorNormG%5VectorScal
arMultiplyG%+ZeroMatrixG%+ZeroVectorG%$ZipG" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 25 "M:=Matrix(2,2,[1,1,2,0]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"MG-%'RTABLEG6%\")!)G)o)-%'MATRIXG6#7$7$\"\"\"F.7$\"
\"#\"\"!%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "v0:=Ma
trix(2,1,[0,1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#v0G-%'RTABLEG6%
\")SUP(*-%'MATRIXG6#7$7#\"\"!7#\"\"\"%'MatrixG" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 103 "# gera os primeiros onze\nfor k from 1 to 11 \+
do\n  # multiplica\347\343o matricial \351 ponto\n  k, M^k.v0;\nend do
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"\"-%'RTABLEG6%\")[%yu*-%'MATR
IXG6#7$7#F#7#\"\"!%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"#-%
'RTABLEG6%\")?'>v*-%'MATRIXG6#7$7#\"\"\"7#F#%'MatrixG" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6$\"\"$-%'RTABLEG6%\")o#zv*-%'MATRIXG6#7$7#F#7#\"\"#
%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"%-%'RTABLEG6%\")7fj(*
-%'MATRIXG6#7$7#\"\"&7#\"\"'%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6$\"\"&-%'RTABLEG6%\")+'3x*-%'MATRIXG6#7$7#\"#67#\"#5%'MatrixG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"'-%'RTABLEG6%\")#Rux*-%'MATRIXG6#7
$7#\"#@7#\"#A%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"(-%'RTAB
LEG6%\")s(=b)-%'MATRIXG6#7$7#\"#V7#\"#U%'MatrixG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$\"\")-%'RTABLEG6%\")kNd')-%'MATRIXG6#7$7#\"#&)7#\"#')%'
MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"*-%'RTABLEG6%\")Scn&)-%
'MATRIXG6#7$7#\"$r\"7#\"$q\"%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6$\"#5-%'RTABLEG6%\")%3qu*-%'MATRIXG6#7$7#\"$T$7#\"$U$%'MatrixG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#6-%'RTABLEG6%\"(WP6%-%'MATRIXG6#7$7
#\"$$o7#\"$#o%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "#
 compara com o resultado do primeiro loop\nA[11],F[11];" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6$\"$$o\"$#o" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 51 "# tentar ingenuamente obter a forma fechada\nM^n.v0;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\".G6$)-%'RTABLEG6%\")!)G)o)-%'M
ATRIXG6#7$7$\"\"\"F07$\"\"#\"\"!%'MatrixG%\"nG-F(6%\")SUP(*-F,6#7$7#F3
7#F0F4" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "Ok, o Maple n\343o entr
ega o jogo de como ele escreveu aquela forma fechada." }}{PARA 0 "" 0 
"" {TEXT -1 97 "Pelo que sabemos de \341lgebra linear, se conseguirmos
 diagonalizar M, ou seja, se pudermos escrever" }}{PARA 0 "" 0 "" 
{XPPEDIT 18 0 "M=S*Lambda*S^(-1)" "6#/%\"MG*(%\"SG\"\"\"%'LambdaGF')F&
,$F'!\"\"F'" }{TEXT -1 6 ", com " }{XPPEDIT 18 0 "Lambda" "6#%'LambdaG
" }{TEXT -1 27 " uma matriz diagonal, ent\343o" }}{PARA 256 "" 0 "" 
{TEXT -1 0 "" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "M^k=S*Lambda^k*S^(-1)
" "6#/)%\"MG%\"kG*(%\"SG\"\"\")%'LambdaGF&F))F(,$F)!\"\"F)" }{TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "e sabemos ainda que as entradas de
 " }{XPPEDIT 18 0 "Lambda" "6#%'LambdaG" }{TEXT -1 71 " s\343o autoval
ores de M e as colunas de S os autovetores correspondentes." }}{PARA 
0 "" 0 "" {TEXT -1 47 "(veja a planilha sobre \341lgebra linear no sit
e!)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "# achar autovetores \+
e autovalores\nEigenvectors(M);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'
RTABLEG6%\")))G*y*-%'MATRIXG6#7$7#\"\"#7#!\"\"&%'VectorG6#%'columnG-F$
6%\")c6*y*-F(6#7$7$\"\"\"#F.F,7$F:F:%'MatrixG" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 9 "(l,S):=%;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>6$%
\"lG%\"SG6$-%'RTABLEG6%\")))G*y*-%'MATRIXG6#7$7#\"\"#7#!\"\"&%'VectorG
6#%'columnG-F)6%\")c6*y*-F-6#7$7$\"\"\"#F3F17$F?F?%'MatrixG" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "# arma matriz diagonal\nLamb
da:=DiagonalMatrix([l[1],l[2]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
'LambdaG-%'RTABLEG6%\")7!or*-%'MATRIXG6#7$7$\"\"#\"\"!7$F/!\"\"%'Matri
xG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "# conferindo\nM=S.Lam
bda.S^(-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'RTABLEG6%\")!)G)o)-
%'MATRIXG6#7$7$\"\"\"F-7$\"\"#\"\"!%'MatrixG-F%6%\")W-4)*F(F1" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Ln:=DiagonalMatrix([l[1]^n,l
[2]^n]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#LnG-%'RTABLEG6%\")7e5)*
-%'MATRIXG6#7$7$)\"\"#%\"nG\"\"!7$F1)!\"\"F0%'MatrixG" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "#finalmente vemos a forma fechada\n
vn:=S.Ln.S^(-1).v0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#vnG-%'RTABLE
G6%\")CP;)*-%'MATRIXG6#7$7#,&*&\"\"$!\"\")\"\"#%\"nG\"\"\"F5*&F0F1)F1F
4F5F17#,&*&F0F1F2F5F5*(F3F5F0F1F7F5F5%'MatrixG" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 70 "Ok, mas como podemos ter certeza que nossa matriz se
r\341 diagonaliz\341vel?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "# nem toda matriz \351 diagonaliz
\341vel\nB:=Matrix(2,2,[1,1,0,1]);\nEigenvectors(B);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%\"BG-%'RTABLEG6%\")gs;)*-%'MATRIXG6#7$7$\"\"\"F.7$
\"\"!F.%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'RTABLEG6%\")S#[
$)*-%'MATRIXG6#7$7#\"\"\"F+&%'VectorG6#%'columnG-F$6%\")3M:')-F(6#7$7$
F,\"\"!7$F8F8%'MatrixG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 128 "De fat
o, a matriz de \"autovetores\" acima tem algo errado. N\343o podemos t
er uma coluna de zeros e ainda ter uma matriz invert\355vel." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "Uns resultados \+
\372teis:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
271 8 "Teorema:" }{TEXT -1 62 " Se M nxn tem n autovalores distintos, \+
ent\343o \351 diagonaliz\341vel." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 272 20 "Teorema (Espectral):" }{TEXT -1 80 " Se \+
M nxn \351 sim\351trica (M=M^T), ent\343o M \351 diagonaliz\341vel e t
em autovalores reais." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "# uma matriz real com autovalores i
magin\341rios\nB:=Matrix(2,2,[0,1,-1,0]);\nEigenvectors(B);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'RTABLEG6%\")7yP)*-%'MATRIXG6#7$7$\"
\"!\"\"\"7$!\"\"F.%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'RTAB
LEG6%\");!o))*-%'MATRIXG6#7$7#^#\"\"\"7#^#!\"\"&%'VectorG6#%'columnG-F
$6%\")w=())*-F(6#7$7$F/F,7$F-F-%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 189 "# mas uma matriz sim\351trica qualquer\nA:=RandomMat
rix(3,3,outputoptions=[shape=symmetric]);\n# confere simetria\nA^%T=A;
\n# e autovalores reais, a menos de erros num\351ricos\nevalf(Eigenval
ues(A));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6%\")o#=*)
*-%'MATRIXG6#7%7%\"#n!#J\"##*7%F/\"#W\"#H7%F0F3\"#**%'MatrixG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'RTABLEG6%\")w\"G*)*-%'MATRIXG6#7%7
%\"#n!#J\"##*7%F.\"#W\"#H7%F/F2\"#**%'MatrixG-F%6%\")o#=*)*F(F5" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")S?.**-%'MATRIXG6#7%7#^$
$\"+:x@k<!\"($!\"%!\")7#^$$!+\">tdN$F2$!+;;5kC!#<7#^$$\"+Rgf8nF2$\"+;;
5kWF9&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 133 "ob
s: os n\372meros complexos acima s\343o puramente erro num\351rico (no
te que a parte imagin\341ria \351 ordens de grandeza menor que a parte
 real)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 
"Maple tem v\341rias fun\347\365es relativas a autovalores/vetores." }
}{PARA 0 "" 0 "" {TEXT -1 55 "Os autovalores s\343o ra\355zes do polin
\364mio caracter\355stico: " }}{PARA 0 "" 0 "" {TEXT -1 25 "p(lambda)=
det(M-lambda*I)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "id:=Iden
tityMatrix(2);\npc:=Determinant(M-lambda*id);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#idG-%'RTABLEG6%\")GI#*)*-%'MATRIXG6#7$7$\"\"\"\"\"!7
$F/F.%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#pcG,(%'lambdaG!\"
\"*$)F&\"\"#\"\"\"F+F*F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 
"# ra\355zes de pc s\343o autovalores\nsolve(pc=0);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6$\"\"#!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
66 "# fun\347\343o para o polin. caract.\nCharacteristicPolynomial(M,l
ambda);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(%'lambdaG!\"\"*$)F$\"\"#
\"\"\"F)F(F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Eigenvalues
(M);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")+8>(*-%'MATRIXG
6#7$7#\"\"#7#!\"\"&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "Exerc\355cio:
" }}{PARA 0 "" 0 "" {TEXT -1 16 "1) resolva para " }{XPPEDIT 18 0 "A[n
]" "6#&%\"AG6#%\"nG" }{TEXT -1 3 " e " }{XPPEDIT 18 0 "F[n]" "6#&%\"FG
6#%\"nG" }{TEXT -1 39 ", mas em vez de em um tri\342ngulo, em um " }
{TEXT 256 9 "tetraedro" }{TEXT -1 72 ". Considere se quiser, um quadra
do com as diagonais para represent\341-lo. " }}{PARA 0 "" 0 "" {TEXT 
-1 24 "2) tente o mesmo com um " }{TEXT 274 9 "pent\341gono" }{TEXT 
-1 14 " com todas as " }{TEXT 275 9 "diagonais" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 11 "3) para um " }{TEXT 273 8 "quadrado" }
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, adjacentes e opostos. Tente obter 3 ED's relacionando " }{XPPEDIT 
18 0 "F[n]" "6#&%\"FG6#%\"nG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "Ad[n]" 
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