{VERSION 2 3 "IBM INTEL NT" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{PSTYLE " Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 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 Plot" 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 }{PSTYLE "Fixed Width" 0 17 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 17 0 } {PSTYLE "" 0 256 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 }{PSTYLE "" 17 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "Example: Given a quadratic equation:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "8*x[1]^2+8*x[2]^2-16*x[1]*x[2]+33*sqrt( 2)-31*sqrt(2)*x[2]+70 = 0;" "/,.*&\"\")\"\"\"*$&%\"xG6#F&\"\"#F&F&*&F% F&*$&F)6#F+F+F&F&*(\"#;F&&F)6#F&F&&F)6#F+F&!\"\"*&\"#LF&-%%sqrtG6#F+F& F&*(\"#JF&-F:6#F+F&&F)6#F+F&F6\"#qF&\"\"!" }{TEXT -1 2 " ," }}{PARA 0 "" 0 "" {TEXT -1 145 "Find the substitutions that transformations equa tion into a standard form of either a parabola, ellipse, or a hyperbol a. Then sketch the graph. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "A:=matrix (2,2,[8,-8,-8,8]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'MATRIXG 6#7$7$\"\")!\")7$F+F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 26 "The program \"eigenmat.txt\"" }}{PARA 257 "" 0 "" {TEXT -1 1436 "\n#\n# program that find eigenvalue matrix \+ and \n# eigenvector matrix of a given symmetric matrix\n#\n# That is: \+ for given symmetric matrix A, the program\n# finds matrix Lambda and X such that\n# \n# A = X*Lambda*(transpose(X))\n#\n# where tran spose(X)*X = I\n#\neigenmat:=proc(\n A::matrix, # input matrix\n Lambda::evaln, # output: eigenvalue matrix\n \+ X::evaln # output: eigenvector matrix\n )\n \+ local n, s, i, j, k, l, m, u, v, lambda;\n\n n:=linalg[rowdim](A); \n if n<> linalg[coldim](A) then\n print(`expecting a square ma trix`);\n RETURN();\n fi;\n\n s:=[linalg[eigenvects](A)];\n \+ m:=nops(s);\n\n X:=matrix(n,n);\n Lambda:=matrix(n,n,[seq(0,i=1.. n*n)]);\n\n if m = n then\n\n for j from 1 to n do\n La mbda[j,j]:=op(1,op(j,s));\n v:=op(1,op(3,op(j,s))); # get the eigenvector\n v:=linalg[normalize](v);\n for i from 1 to n do\n X[i,j]:=v[i]\n od;\n od;\n\n else \n\n for j from 1 to m do\n k:=op(2,op(j,s)); # ge t multiplicity\n lambda:=op(1,op(j,s)); # get eigenvalue\n \+ u:=op(3,op(j,s)); # get eigenvector set\n for l \+ from 1 to k do\n Lambda[j+l-1, j+l-1]:=lambda;\n \+ v:=linalg[normalize](op(l,u));\n for i from 1 to n do\n \+ X[i,j+l-1]:=v[i]\n od;\n od;\n od; \n\n fi;\n\n print(evalm(Lambda),evalm(X));\n\nend;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "read(`a:eigenmat.txt`);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%)eigenmatG:6%'%\"AG%'matrixG'%'LambdaG%&ev alnG'%\"XGF,6,%\"nG%\"sG%\"iG%\"jG%\"kG%\"lG%\"mG%\"uG%\"vG%'lambdaG6 \"F:C*>8$-&%'linalgG6#%'rowdimG6#9$@$0F=-&F@6#%'coldimGFCC$-%&printG6# %:expecting~a~square~matrixG-%'RETURNGF:>8%7#-&F@6#%+eigenvectsGFC>8*- %%nopsG6#FS>9&-F)6$F=F=>9%-F)6%F=F=7#-%$seqG6$\"\"!/8&;\"\"\"*$F=\"\"# @%/FZF=?(8'FhoFhoF=%%trueGC&>&F]o6$F^pF^p-%#opG6$Fho-Fep6$F^pFS>8,-Fep 6$Fho-Fep6$\"\"$Fgp>Fjp-&F@6#%*normalizeG6#Fjp?(FfoFhoFhoF=F_p>&Fin6$F foF^p&Fjp6#Ffo?(F^pFhoFhoFZF_pC&>8(-Fep6$FjoFgp>8-Fdp>8+F]q?(8)FhoFhoF _rF_pC%>&F]o6$,(F^pFhoFgrFho!\"\"FhoF\\sFcr>Fjp-Fbq6#-Fep6$FgrFer?(Ffo FhoFhoF=F_p>&Fin6$FfoF\\sFjq-FM6$-%&evalmG6#F]o-Fjs6#FinF:F:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "eigenmat(A,Lambda,X);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$-%'MATRIXG6#7$7$\"#;\"\"!7$F)F)-F$6#7$ 7$,$*$\"\"##\"\"\"F1#!\"\"F1,$F0F27$F6F6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "evalm(Lambda);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%' MATRIXG6#7$7$\"#;\"\"!7$F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalm(X);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7$7$,$*$ \"\"##\"\"\"F*#!\"\"F*,$F)F+7$F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "inverse(X);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MAT RIXG6#7$7$,$*$\"\"##\"\"\"F*#!\"\"F*,$F)F+7$F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "b:=vector([33*sqrt(2),-31*sqrt(2)]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG-%'VECTORG6#7$,$*$\"\"##\"\"\"F+ \"#L,$F*!#J" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "multiply(tra nspose(b),X);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*transposeG6#-%'VEC TORG6#7$!#k\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "y:=vect or(2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yG-%&arrayG6$;\"\"\"\"\" #7\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "multiply(transpose( y),Lambda,y)+multiply(transpose(b),X,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$&%\"yG6#\"\"\"\"\"#\"#;F%!#k&F&6#F)F)" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 22 "z:=vector([t,-8*t^2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"zG-%'VECTORG6#7$%\"tG,$*$F)\"\"#!\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "plot([z[1],z[2],t=-0.5..0.5]);" }} {PARA 13 "" 1 "" {INLPLOT "6#-%'CURVESG6$7S7$$!1+++++++]!#;$!\"#\"\"!7 $$!1LLLe%G?y%F*$!1!z/TpB%H=!#:7$$!1mmT&esBf%F*$!1CH1x3>(o\"F37$$!1LL$3 s%3zVF*$!1w)y$R16M:F37$$!1LL$e/$QkTF*$!1HR>#*oO(Q\"F37$$!1nmT5=q]RF*$! 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