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"" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 258 "" 0 "" {TEXT 256 7 "Systems" }}{PARA 19 "" 0 "" {TEXT 258 15 "5.1-5.8epR4.mws" }}{PARA 259 "" 0 "" {TEXT 257 14 " Brian E. Blank" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 " " 0 "" {TEXT -1 84 "Click on a [+] sign to expand a section. Click on a [-] sign to collapse a section." }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 8 "Keywords" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{HYPERLNK 17 "addrow " 2 "linalg[addrow]" "" }{TEXT -1 5 ", " }{HYPERLNK 17 "charpoly" 2 "linalg[charpoly]" "" }{TEXT -1 4 ", " }{HYPERLNK 17 "det" 2 "det " "" }{TEXT -1 5 ", " }{HYPERLNK 17 "eigenvals" 2 "eigenvects" "" } {TEXT -1 6 ", " }{HYPERLNK 17 "eigenvects" 2 "linalg[eigenvects]" "" }{TEXT -1 4 ", " }{HYPERLNK 17 "evalm" 2 "evalm" "" }{TEXT -1 4 " , " }{HYPERLNK 17 "linalg" 2 "linalg" "" }{TEXT -1 5 ", " } {HYPERLNK 17 "linsolve" 2 "linsolve" "" }{TEXT -1 4 ", " }{HYPERLNK 17 "matrix" 2 "matrix" "" }{TEXT -1 4 ", " }{HYPERLNK 17 "mulrow" 2 "mulrow" "" }{TEXT -1 5 ", " }{HYPERLNK 17 "rref" 2 "linalg[rref]" "" }{TEXT -1 5 ", " }{HYPERLNK 17 "swaprow" 2 "linalg[swaprow]" "" }{TEXT -1 5 ", " }{HYPERLNK 17 "vector" 2 "vector" "" }{TEXT -1 1 " ," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 49 "The Basic Data Types of Linear Algebra I: Vectors" }}{PARA 0 " " 0 "" {TEXT -1 136 "Start by loading the linear algebra package. We w ill use the noisy terminator to get an appreciation of the commands th at are available:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg);" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definition for norm" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warn ing, new definition for trace" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7^r%. BlockDiagonalG%,GramSchmidtG%,JordanBlockG%)LUdecompG%)QRdecompG%*Wron skianG%'addcolG%'addrowG%$adjG%(adjointG%&angleG%(augmentG%(backsubG%% bandG%&basisG%'bezoutG%,blockmatrixG%(charmatG%)charpolyG%)choleskyG%$ colG%'coldimG%)colspaceG%(colspanG%*companionG%'concatG%%condG%)copyin toG%*crossprodG%%curlG%)definiteG%(delcolsG%(delrowsG%$detG%%diagG%(di vergeG%(dotprodG%*eigenvalsG%,eigenvaluesG%-eigenvectorsG%+eigenvectsG %,entermatrixG%&equalG%,exponentialG%'extendG%,ffgausselimG%*fibonacci G%+forwardsubG%*frobeniusG%*gausselimG%*gaussjordG%(geneqnsG%*genmatri xG%%gradG%)hadamardG%(hermiteG%(hessianG%(hilbertG%+htransposeG%)iherm iteG%*indexfuncG%*innerprodG%)intbasisG%(inverseG%'ismithG%*issimilarG %'iszeroG%)jacobianG%'jordanG%'kernelG%*laplacianG%*leastsqrsG%)linsol veG%'mataddG%'matrixG%&minorG%(minpolyG%'mulcolG%'mulrowG%)multiplyG%% normG%*normalizeG%*nullspaceG%'orthogG%*permanentG%&pivotG%*potentialG %+randmatrixG%+randvectorG%%rankG%(ratformG%$rowG%'rowdimG%)rowspaceG% (rowspanG%%rrefG%*scalarmulG%-singularvalsG%&smithG%&stackG%*submatrix G%*subvectorG%)sumbasisG%(swapcolG%(swaprowG%*sylvesterG%)toeplitzG%&t raceG%*transposeG%,vandermondeG%*vecpotentG%(vectdimG%'vectorG%*wronsk ianG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 " As you see, " }{TEXT 305 5 "MAPLE" }{TEXT -1 101 " has a rich supply of functions for dealing with linear algebra. We have already seen th e command " }{TEXT 306 9 "wronskian" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 "We specify a matrix by creating a list of its rows. Each row is a list of entries. For examp le:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "A typica l vector is declared as follows:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "v := vector([a, b, c, d]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG-%'VECTORG6#7&%\"aG%\"bG%\"cG% \"dG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 " Select an entry as follows:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "v[3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"cG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "You can define a general vector by specifying its dimension:" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "w := vector(5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"wG-%&arra yG6$;\"\"\"\"\"&7\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 39 "You can select entries in the same way:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "w[4];" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#&%\"wG6#\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "Various vector operations are available:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 15 "u := vector(5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG-%&arrayG6$;\"\"\"\"\"&7\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "dotprod(u,w);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*& &%\"uG6#\"\"\"F(&%\"wGF'F(F(*&&F&6#\"\"#F(&F*F-F(F(*&&F&6#\"\"$F(&F*F2 F(F(*&&F&6#\"\"%F(&F*F7F(F(*&&F&6#\"\"&F(&F*F " 0 "" {MPLTEXT 1 0 36 "f := (x,y,z) -> sin(x*y^2)*exp(x/z);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"fG:6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowG F**&-%$sinG6#*&9$\"\"\"9%\"\"#F4-%$expG6#*&F3F49&!\"\"F4F*F*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "U := grad(f(x,y,z), vector([ x,y,z]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"UG-%'VECTORG6#7%,&*(- %$cosG6#*&%\"xG\"\"\"%\"yG\"\"#F0F1F2-%$expG6#*&F/F0%\"zG!\"\"F0F0*(-% $sinGF-F0F7F8F3F0F0,$**F+F0F/F0F1F0F3F0F2,$**F:F0F/F0F7!\"#F3F0F8" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "diverge(U, vector([x,y,z])); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,0*(-%$sinG6#*&%\"xG\"\"\"%\"yG\" \"#F*F+\"\"%-%$expG6#*&F)F*%\"zG!\"\"F*F3**-%$cosGF'F*F+F,F2F3F.F*F,*( F%F*F2!\"#F.F*F***F%F*F)F,F+F,F.F*!\"%*(F5F*F)F*F.F*F,**F%F*F)F*F2!\"$ F.F*F,**F%F*F)F,F2F:F.F*F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "It is usefule to have the zero vector of a give n dimension:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 254 "zeroVector := proc(n::posint)\n loca l gooseEggs, j:\n gooseEggs := []:\n for j f rom 1 to n do\n gooseEggs := [op(gooseEggs), 0]:\n \+ od:\n RETURN(vector(gooseEggs));\n en d;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%+zeroVectorG:6#'%\"nG%'posintG 6$%*gooseEggsG%\"jG6\"F-C%>8$7\"?(8%\"\"\"F49$%%trueG>F07$-%#opG6#F0\" \"!-%'RETURNG6#-%'vectorGF;F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "zeroVector(3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6# 7%\"\"!F'F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 51 "The Basic Data Types o f Linear Algebra II: Matrices" }}{PARA 0 "" 0 "" {TEXT -1 47 "\nRestar t and reload the linear algebra package:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "restart:\nwith(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definition for norm" }}{PARA 7 "" 1 "" {TEXT -1 33 "W arning, new definition for trace" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 "We specif y a matrix by creating a list of its rows. Each row is a list of entri es. For example:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "M := matrix( [ [2, 1, 3], [4, 1, 1], [3, 3, 1] ] );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG-%'MATRIXG6#7%7%\"\"#\"\"\"\"\"$7%\"\"%F+F+7%F,F,F+" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Notice th at the syntax is " }{TEXT 259 32 "matrix( [ list1, list2, ... ] )" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "We can augment matrix " }{TEXT 260 1 "M" }{TEXT -1 20 " \+ by column vector " }{TEXT 261 16 " b = [5, -1, 2] " }{TEXT -1 12 " a s follows:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 24 "b := vector([5, -1, 2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG-%'VECTORG6#7%\"\"&!\"\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "M1 := augment(M,b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#M1G-%'MATRIXG6#7%7&\"\"#\"\"\"\"\"$\"\"&7&\"\"%F+F+! \"\"7&F,F,F+F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "We can perform row reduction by means of the commands " } {TEXT 262 7 " addrow" }{TEXT -1 4 ", " }{TEXT 263 6 "mulrow" }{TEXT -1 9 ", and " }{TEXT 264 7 "swaprow" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "M2 := mulrow(M1, 1, 1/2); # multiplies the f irst row of M1 by 1/2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#M2G-%'MATR IXG6#7%7&\"\"\"#F*\"\"##\"\"$F,#\"\"&F,7&\"\"%F*F*!\"\"7&F.F.F*F," }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "M3 := addrow(M2, 1, 2, -4); \+ # adds -4 times row 1 to row 2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#M 3G-%'MATRIXG6#7%7&\"\"\"#F*\"\"##\"\"$F,#\"\"&F,7&\"\"!!\"\"!\"&!#67&F .F.F*F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "M4 := addrow(M3, 1, 3, -3); # adds -3 times row 1 to row 3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#M4G-%'MATRIXG6#7%7&\"\"\"#F*\"\"##\"\"$F,#\"\"&F,7& \"\"!!\"\"!\"&!#67&F2F-#!\"(F,#F5F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "M5 := mulrow(M4, 2, -1); # multiplies row 2 by -1" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#M5G-%'MATRIXG6#7%7&\"\"\"#F*\"\"## \"\"$F,#\"\"&F,7&\"\"!F*F0\"#67&F2F-#!\"(F,#!#6F," }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 62 "M6 := addrow(M5, 2, 1, -1/2); # adds -1/2 ti mes row 2 to row 1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#M6G-%'MATRIXG 6#7%7&\"\"\"\"\"!!\"\"!\"$7&F+F*\"\"&\"#67&F+#\"\"$\"\"##!\"(F4#!#6F4 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "M7 := addrow(M6, 2, 3, \+ -3/2); # adds -3/2 times row 2 to row 3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#M7G-%'MATRIXG6#7%7&\"\"\"\"\"!!\"\"!\"$7&F+F*\"\"&\"#67&F+F+! #6!#A" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "M8 := mulrow(M7, 3 , -1/11); # multiplies row 3 by -1/11" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#M8G-%'MATRIXG6#7%7&\"\"\"\"\"!!\"\"!\"$7&F+F*\"\"&\"#67&F+F+F *\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "M9 := addrow(M8, \+ 3, 1, 1); # adds 1 times row 3 to row 1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#M9G-%'MATRIXG6#7%7&\"\"\"\"\"!F+!\"\"7&F+F*\"\"&\"#67&F+F+F* \"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "M10 := addrow(M9, \+ 3, 2, -5); # adds -5 times row 3 to row 2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$M10G-%'MATRIXG6#7%7&\"\"\"\"\"!F+!\"\"7&F+F*F+F*7&F+ F+F*\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "Incidentally, " }{TEXT 275 5 "MAPLE" }{TEXT -1 50 " does not always display the entries of a matri x." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "M;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"MG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "There are two ways to coax the entries out of it:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalm(M);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%'MATRIXG6#7%7%\"\"#\"\"\"\"\"$7%\"\"%F)F)7%F*F*F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "print(M);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%'MATRIXG6#7%7%\"\"#\"\"\"\"\"$7%\"\"%F)F)7%F*F*F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "Creati ng the identity matrix can be done as follows:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "array(identi ty,1..5,1..5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7'7'\"\" \"\"\"!F)F)F)7'F)F(F)F)F)7'F)F)F(F)F)7'F)F)F)F(F)7'F)F)F)F)F(" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 22 "Operations on Matrices" }}{PARA 0 "" 0 " " {TEXT -1 39 "Matrices can be multiplied by a scalar:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "c*M;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"cG\"\"\"%\"MGF%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalm(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7%,$%\"cG\"\"#F),$F)\"\"$7%,$F)\"\"%F)F)7%F+F+F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "Anothe r way to do this is by means of a command" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "scalarmul(M, c);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7%,$%\"cG\"\"#F),$F)\"\" $7%,$F)\"\"%F)F)7%F+F+F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 32 "Same size matrices can be added:" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "A := matri x([[1,2,3],[4,5,6]]); \nB := matrix([[7,8,9],[10,11,12]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'MATRIXG6#7$7%\"\"\"\"\"#\"\"$7%\"\" %\"\"&\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'MATRIXG6#7$7% \"\"(\"\")\"\"*7%\"#5\"#6\"#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "A + B; \nevalm(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"AG\" \"\"%\"BGF%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7$7%\"\")\" #5\"#77%\"#9\"#;\"#=" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 87 "Matrices of appropriate sizes can be multiplied. There are two different ways to do so." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "C := matrix([[2,0,1,-1],[4,5 ,6,1],[2,3,4,5]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CG-%'MATRIXG 6#7%7&\"\"#\"\"!\"\"\"!\"\"7&\"\"%\"\"&\"\"'F,7&F*\"\"$F/F0" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "multiply(A, C);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7$7&\"#;\"#>\"#DF(7&\"#S\"#V\"#e\"# J" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 "or" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "A &* C; \nevalm(\");" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%#&*G6$%\"AG%\"CG" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%'MATRIXG6#7$7&\"#;\"#>\"#DF(7&\"#S\"#V\"#e\"#J" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "However, \+ just using the multiplication operator does not work:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "A*C;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"AG\"\"\"%\"CGF%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalm(\");" }}{PARA 8 "" 1 "" {TEXT -1 79 "Error, (in evalm/evaluate) use the &* operator for matrix/vector mult iplication" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "If the determinant of a square matrix is not zero, we can inver t it:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "det(M);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#A" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "i nverse(M);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7%#!\"\"\" #6#\"\"%F*F(7%#F)\"#A#!\"(F/#\"\"&F*7%#\"\"*F/#!\"$F/F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "multiply(M, inverse(M));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7%\"\"\"\"\"!F)7%F)F(F)7%F)F)F (" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 35 " Solving Systems of Linear Equations" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 40 "Consider the system of linear equations: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "eqn1 := 2*x + y + 3*z = 5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%%eqn1G/,(%\"xG\"\"#%\"yG\"\"\"%\"zG\"\"$\"\"&" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 26 "eqn2 := 4*x + y + z = -1;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%%eqn2G/,(%\"xG\"\"%%\"yG\"\"\"%\"zGF*!\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "eqn3 := 3*x + 3*y + z = 2; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn3G/,(%\"xG\"\"$%\"yGF(%\"zG \"\"\"\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "There are many ways to solve this system of equations." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 265 10 "I) Using " }{TEXT 266 5 "solve" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "eqn_set := \{ eqn1, eqn2, eqn3 \};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(eqn_ setG<%/,(%\"xG\"\"%%\"yG\"\"\"%\"zGF+!\"\"/,(F(\"\"$F*F0F,F+\"\"#/,(F( F1F*F+F,F0\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "solve(eq n_set, \{x, y, z\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/%\"xG!\"\"/ %\"yG\"\"\"/%\"zG\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 267 12 "II) Using " }{TEXT 272 8 "linsolve" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 "With the linear alge bra package loaded, we do not need to type the variables. We call on \+ " }{TEXT 268 8 "linsolve" }{TEXT -1 149 " with the coefficient matrix as the first argument and the vector right hand side as the second. I n this case the coefficient matrix is the matrix " }{TEXT 269 1 "M" } {TEXT -1 66 " we have been row reducing and the right hand side is the vector " }{TEXT 270 1 "b" }{TEXT -1 38 ". The answer is returned as a vector." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 14 "linsolve(M,b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#- %'VECTORG6#7%!\"\"\"\"\"\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 271 24 "III) Using row reduction" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 137 "We have already don e the row reduction. The answer we seek is the last column of the matr ix that arose in the last step of row reduction." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "col(M10, 4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-% 'VECTORG6#7%!\"\"\"\"\"\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 84 "Of course provides a one-step command th at will do the row reduction automatically:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "rref(M1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIX G6#7%7&\"\"\"\"\"!F)!\"\"7&F)F(F)F(7&F)F)F(\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "This command, " } {MPLTEXT 1 0 4 "rref" }{TEXT -1 71 ", stands for the technical termino logy for the result of the process: " }{TEXT 273 24 "row reduced eche lon form" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 276 10 "IV) Using " }{TEXT 277 9 " inverse " }}{PARA 0 " " 0 "" {TEXT -1 15 "If the matrix " }{MPLTEXT 1 0 1 "M" }{TEXT -1 64 " is invertible then there is an explicit form of the solution " } {MPLTEXT 1 0 10 "M &* x = b" }{TEXT -1 9 ", namely " }{MPLTEXT 1 0 20 " x = inverse(M) &* b" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "evalm(inverse(M) &* b); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%!\"\"\"\"\"\"\"#" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "Notice t hat " }{MPLTEXT 1 0 2 "&*" }{TEXT -1 57 " is the operator for matrix multiplication: the simple " }{MPLTEXT 1 0 1 "*" }{TEXT -1 15 " wi ll not do." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 " " {TEXT -1 28 "Eigenvalues and Eigenvectors" }}{PARA 0 "" 0 "" {TEXT -1 50 "There are many ways to determine eigenvalues in " }{TEXT 274 5 "MAPLE" }{TEXT -1 56 " . There is a command for the characteristic p olynomial." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 55 "A := matrix([[11, 6, 30], [-3, 2, -12], [-3, -3, -7 ]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'MATRIXG6#7%7%\"#6\"\" '\"#I7%!\"$\"\"#!#77%F.F.!\"(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "charpoly(A, lambda);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**$%'la mbdaG\"\"$\"\"\"*$F%\"\"#!\"'F%F&\"#5F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "We can get the eigenvalues by solv ing this equation:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "solve(\", la mbda);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"&\"\"#!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "There is also a o ne-step command:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 20 "r := [eigenvals(A)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG7%\"\"&\"\"#!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "We can solve for eigenvectors the lo ng way:" }}{PARA 0 "" 0 "" {TEXT -1 36 "First we create the identity m atrix:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Id := array(identity, 1..3,1..3);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#IdG-%&arrayG6&%)identityG;\"\"\"\"\"$F)7\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "A - r[1]*Id;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"AG\"\"\"%#IdG!\"&" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 9 "evalm(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MAT RIXG6#7%7%\"\"'F(\"#I7%!\"$F+!#7F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "rref(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIX G6#7%7%\"\"\"F(\"\"!7%F)F)F(7%F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 140 "From this we see that the first two \+ components of an eigenvector corresponding to 5 must sum to zero. The \+ third component must be zero. Thus" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "v[1] := vector([1,-1,0]);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"vG6#\"\"\"-%'VECTORG6#7%F'!\"\" \"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "is an eigenvector. As a check:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "evalm( A &* v[1] - r[1]*v[1] );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #-%'VECTORG6#7%\"\"!F'F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 87 "In a similar way we can find eigenvectors that cor respond to the other two eigenvalues." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 77 "There is also a one-step command to g et all eigenvalues and all eigenvectors:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "eigenvects(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%7%! \"\"\"\"\"<#-%'VECTORG6#7%!\"$F%F%7%\"\"#F%<#-F(6#7%!\"%F%F%7%\"\"&F%< #-F(6#7%F%F$\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 165 "The first number in each triple is the eigenvalue. The n ext number is its multiplicity. The third entry of each triple is a se t of eigenvectors for that eigenvalue. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 69 "Solving a First Order Linear System Case I: Full set of Eigenve ctors " }}{SECT 1 {PARA 4 "" 0 "" {TEXT 278 9 "Example 1" }}{PARA 0 " " 0 "" {TEXT -1 31 "Solve the initial value problem" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "ode[1] := diff(x(t), t) = 11*x(t) + 6*y(t) + 30* z(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$odeG6#\"\"\"/-%%diffG6$-% \"xG6#%\"tGF/,(F,\"#6-%\"yGF.\"\"'-%\"zGF.\"#I" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 53 "ode[2] := diff(y(t), t) = -3*x(t) + 2*y(t) - 1 2*z(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$odeG6#\"\"#/-%%diffG6$- %\"yG6#%\"tGF/,(-%\"xGF.!\"$F,F'-%\"zGF.!#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "ode[3] := diff(z(t), t) = -3*x(t) - 3*y(t) - 7*z(t );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$odeG6#\"\"$/-%%diffG6$-%\"zG 6#%\"tGF/,(-%\"xGF.!\"$-%\"yGF.F3F,!\"(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "iv[1] := x(0) = 5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>&%#ivG6#\"\"\"/-%\"xG6#\"\"!\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "iv[2] := y(0) = 1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>&%#ivG6#\"\"#/-%\"yG6#\"\"!\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "iv[3] := z(0) = -4;" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>&%#ivG6#\"\"$/-%\"zG6#\"\"!!\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 25 "The coefficient matrix is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "A := matrix([[11, 6, 30], [-3, 2, -12], [-3, \+ -3, -7]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'MATRIXG6#7%7%\" #6\"\"'\"#I7%!\"$\"\"#!#77%F.F.!\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "with eigenvectors" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "eigenvects(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6% 7%\"\"&\"\"\"<#-%'VECTORG6#7%!\"\"F%\"\"!7%\"\"#F%<#-F(6#7%!\"%F%F%7%F +F%<#-F(6#7%!\"$F%F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "Therefore the general solution is of the form" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "c[1]*exp(5*t)*vector([-1, 1, 0]) + c[2]*exp(2*t)*vector([-4, 1, 1 ]) + c[3]*exp(-t)*vector([-3, 1, 1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*(&%\"cG6#\"\"\"F(-%$expG6#,$%\"tG\"\"&F(-%'VECTORG6#7%!\"\"F(\" \"!F(F(*(&F&6#\"\"#F(-F*6#,$F-F8F(-F06#7%!\"%F(F(F(F(*(&F&6#\"\"$F(-F* 6#,$F-F3F(-F06#7%!\"$F(F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "augment(vector([-1, 1, 0]), vector([-4, 1, 1]), vector([-3, 1, 1 ]), vector([5, 1, -4]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG 6#7%7&!\"\"!\"%!\"$\"\"&7&\"\"\"F-F-F-7&\"\"!F-F-F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "rref(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7&\"\"\"\"\"!F)\"\"&7&F)F(F)\"\"#7&F)F)F(!\"'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 122 "vector([x(t), y(t), z(t)]) \+ = 5*exp(5*t)*vector([-1, 1, 0]) + 2*exp(2*t)*vector([-4, 1, 1]) - 6*ex p(-t)*vector([-3, 1, 1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'VECTO RG6#7%-%\"xG6#%\"tG-%\"yGF*-%\"zGF*,(*&-%$expG6#,$F+\"\"&\"\"\"-F%6#7% !\"\"F7\"\"!F7F6*&-F36#,$F+\"\"#F7-F%6#7%!\"%F7F7F7FA*&-F36#,$F+F;F7-F %6#7%!\"$F7F7F7!\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "soln := map( u -> evalm(u) , \" );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%s olnG/-%'VECTORG6#7%-%\"xG6#%\"tG-%\"yGF,-%\"zGF,-F'6#7%,(-%$expG6#,$F- \"\"&!\"&-F76#,$F-\"\"#!\")-F76#,$F-!\"\"\"#=,(F6F:F " 0 "" {MPLTEXT 1 0 116 "soln_list := \{\}:\nfor i f rom 1 to 3 do\n s[i] := lhs(soln)[i] = rhs(soln)[i];\n soln_list : = \{op(soln_list),\"\}:\nod;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"s G6#\"\"\"/-%\"xG6#%\"tG,(-%$expG6#,$F,\"\"&!\"&-F/6#,$F,\"\"#!\")-F/6# ,$F,!\"\"\"#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*soln_listG<#/-%\"x G6#%\"tG,(-%$expG6#,$F*\"\"&!\"&-F-6#,$F*\"\"#!\")-F-6#,$F*!\"\"\"#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"sG6#\"\"#/-%\"yG6#%\"tG,(-%$exp G6#,$F,\"\"&F2-F/6#,$F,F'F'-F/6#,$F,!\"\"!\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*soln_listG<$/-%\"xG6#%\"tG,(-%$expG6#,$F*\"\"&!\"&-F -6#,$F*\"\"#!\")-F-6#,$F*!\"\"\"#=/-%\"yGF),(F,F0F2F5F7!\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"sG6#\"\"$/-%\"zG6#%\"tG,&-%$expG6#,$F, \"\"#F2-F/6#,$F,!\"\"!\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*soln_l istG<%/-%\"xG6#%\"tG,(-%$expG6#,$F*\"\"&!\"&-F-6#,$F*\"\"#!\")-F-6#,$F *!\"\"\"#=/-%\"yGF),(F,F0F2F5F7!\"'/-%\"zGF),&F2F5F7F@" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "for i from 1 to 3\ndo simplify(sub s(t = 0, soln_list[i])):\nrhs(\")-rhs(iv[i]);\nod;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%\"xG6#\"\"!\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#\"\"!\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"zG6#\"\"!!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "Another w ay to see that our solution is correct is to compare it with the black box solution:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "dsolve(\{ode[1],ode[2],ode[3],iv[1],iv[2],iv[3] \},\{x(t), y(t), z(t)\} );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/-%\"x G6#%\"tG,(-%$expG6#,$F(\"\"&!\"&-F+6#,$F(\"\"#!\")-F+6#,$F(!\"\"\"#=/- %\"yGF',(F*F.F0F3F5!\"'/-%\"zGF',&F0F3F5F>" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 279 9 "Example 2" }}{PARA 0 "" 0 " " {TEXT -1 31 "Solve the initial value problem" }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{XPPEDIT 280 1 "diff(x(t),t) = 17*x(t)+20*y(t)+40*z(t ),diff(y(t),t) = -5*x(t)-8*y(t)+10*z(t),diff(z(t),t) = -5*x(t)-5*y(t) \+ -13*z(t)" "6%/-%%diffG6$-%\"xG6#%\"tGF*,(*&\"#<\"\"\"-F(6#F*F.F.*&\"#? F.-%\"yG6#F*F.F.*&\"#SF.-%\"zG6#F*F.F./-F%6$-F46#F*F*,(*&\"\"&F.-F(6#F *F.!\"\"*&\"\")F.-F46#F*F.FE*&\"#5F.-F96#F*F.F./-F%6$-F96#F*F*,(*&FBF. -F(6#F*F.FE*&FBF.-F46#F*F.FE*&\"#8F.-F96#F*F.FE" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "with init ial values" }{TEXT 281 4 " " }{XPPEDIT 282 1 "x(0) = 2,y(0) = 1,z(0 ) = 7" "6%/-%\"xG6#\"\"!\"\"#/-%\"yG6#F'\"\"\"/-%\"zG6#F'\"\"(" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "ode[1] := di ff(x(t), t) = 17*x(t) + 20*y(t) + 40*z(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$odeG6#\"\"\"/-%%diffG6$-%\"xG6#%\"tGF/,(F,\"#<-%\"y GF.\"#?-%\"zGF.\"#S" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "ode[ 2] := diff(y(t), t) = - 5*x(t) - 8*y(t) + 10*z(t);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>&%$odeG6#\"\"#/-%%diffG6$-%\"yG6#%\"tGF/,(-%\"xGF.! \"&F,!\")-%\"zGF.\"#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "od e[3] := diff(z(t), t) = -5*x(t) - 5*y(t) - 13*z(t);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>&%$odeG6#\"\"$/-%%diffG6$-%\"zG6#%\"tGF/,(-%\"xGF.! \"&-%\"yGF.F3F,!#8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "iv[1] := x(0) = 2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#ivG6#\"\"\"/-%\"x G6#\"\"!\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "iv[2] := y (0) = 1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#ivG6#\"\"#/-%\"yG6#\" \"!\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "iv[3] := z(0) \+ = 7;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#ivG6#\"\"$/-%\"zG6#\"\"!\" \"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "T he coefficient matrix is\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "A :=matrix([[17, 20, 40], [-5, -8, -10], [-5, -5, -13]]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'MATRIXG6#7%7%\"#<\"#?\"#S7%! \"&!\")!#57%F.F.!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "with characteristic pol ynomial" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "charpoly(A, lambda);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#,**$%'lambdaG\"\"$\"\"\"*$F%\"\"#\"\" %F%!\"$!#=F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "and eigenvalues" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "solve(\");" }}{PARA 11 "" 1 " " {XPPMATH 20 "6%\"\"#!\"$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 49 "Let us find an eigenvector that correspon ds to 2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "evalm(A - lambda*Id);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7%,&\"#<\"\"\"%'lambdaG!\"\"\"#?\"#S7%!\"&,&!\")F* F+F,!#57%F0F0,&!#8F*F+F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "subs(lambda = 2, \" );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6 #7%7%\"#:\"#?\"#S7%!\"&!#5F-7%F,F,!#:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "mulrow(\", 3, -1/5); # multiplies row 3 by -1/5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7%\"#:\"#?\"#S7%!\"&!#5F -7%\"\"\"F/\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "addrow( \", 3, 1, -15); # adds -15 times row 3 to row 1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7%\"\"!\"\"&!\"&7%F*!#5F,7%\"\"\"F.\"\"$ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "addrow(\", 3, 2, 5); # \+ adds 5 times row 3 to row 2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATR IXG6#7%7%\"\"!\"\"&!\"&7%F(F*F)7%\"\"\"F-\"\"$" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 46 "swaprow(\", 1, 3); # interchanges rows 1 and \+ 3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7%\"\"\"F(\"\"$7%\" \"!!\"&\"\"&7%F+F-F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "mul row(\", 2, -1/5); # multiplies row 2 by -1/5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7%\"\"\"F(\"\"$7%\"\"!F(!\"\"7%F+\"\"&!\" &" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "addrow(\", 2, 1, -1); \+ # adds -1 times row 2 to row 1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%' MATRIXG6#7%7%\"\"\"\"\"!\"\"%7%F)F(!\"\"7%F)\"\"&!\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "addrow(\", 2, 3, -5); # adds -5 ti mes row 2 to row 3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7% \"\"\"\"\"!\"\"%7%F)F(!\"\"7%F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "This tells us that if " }{TEXT 291 18 "vector( [a, b, c])" }{TEXT -1 48 " is an eigenvector correspondi ng to -3 then " }{TEXT 292 11 "a + 4*c = 0" }{TEXT -1 10 ". and \+ " }{TEXT 296 9 "b - c = 0" }{TEXT -1 92 ". With two equations among th ree unknown we can assign one variable arbitrarily. If we set " } {TEXT 293 9 " c = -1 " }{TEXT -1 10 " then " }{TEXT 294 5 "a = 4 " }{TEXT -1 7 " and " }{TEXT 295 7 " b = -1" }{TEXT -1 9 ". We get " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "c[1]*vector([4, -1, -1])*exp(2*t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(&%\"cG6#\"\"\"F'-%'VECTORG6#7%\"\"%!\"\"F-F'-%$expG6# ,$%\"tG\"\"#F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "as the summand of the general solution that corresponds t o eigenvalue 2. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Let us turn to the remining eigenvalue." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 185 "Notice that -3 occurs \+ with multiplicity 2. Until we find the space of eigenvectors that cor responds to this eigenvalue we will not know if this higher multiplici ty causes any troubles." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "evalm (A - lambda*Id);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7%,& \"#<\"\"\"%'lambdaG!\"\"\"#?\"#S7%!\"&,&!\")F*F+F,!#57%F0F0,&!#8F*F+F, " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "subs(lambda = -3, \" ); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7%\"#?F(\"#S7%!\"&F+ !#5F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "mulrow(\", 1, 1/20 ); # multiply row 1 by 1/20" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATR IXG6#7%7%\"\"\"F(\"\"#7%!\"&F+!#5F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "addrow(\", 1, 2, 5); # add five times row 1 to row 2 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7%\"\"\"F(\"\"#7%\" \"!F+F+7%!\"&F-!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "addro w(\", 1, 3, 5); # add five times row 1 to row 3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7%\"\"\"F(\"\"#7%\"\"!F+F+F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "This tells us tha t if " }{TEXT 283 18 "vector( [a, b, c])" }{TEXT -1 48 " is an eig envector corresponding to -3 then " }{TEXT 284 15 "a + b + 2*c = 0" }{TEXT -1 177 ". With one equation among three unknown we can assign t wo variables arbitrarily. That tells us that we can find two independe nt eigenvectors corresponding to -3. For one, set " }{TEXT 285 6 "a \+ = 2" }{TEXT -1 8 " and " }{TEXT 286 5 "b = 0" }{TEXT -1 8 ". Then \+ " }{TEXT 287 7 " c = -1" }{TEXT -1 23 ". For the other, set " } {TEXT 288 6 "a = 0" }{TEXT -1 8 " and " }{TEXT 289 5 "b = 2" } {TEXT -1 8 ". Then " }{TEXT 290 7 " c = -1" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "We get" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "c[ 2]*vector( [2, 0, -1])*exp(-3*t) + c[3]*vector( [0, 2, -1])*exp(-3*t); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*(&%\"cG6#\"\"#\"\"\"-%'VECTORG6 #7%F(\"\"!!\"\"F)-%$expG6#,$%\"tG!\"$F)F)*(&F&6#\"\"$F)-F+6#7%F.F(F/F) F0F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "as the part of the general solution that corresponds to eigenvalue -3. Thus" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "c[1]*vector([4, -1, \+ -1])*exp(2*t) + c[2]*vector( [2, 0, -1])*exp(-3*t) + c[3]*vector( [0, \+ 2, -1])*exp(-3*t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*(&%\"cG6#\"\" \"F(-%'VECTORG6#7%\"\"%!\"\"F.F(-%$expG6#,$%\"tG\"\"#F(F(*(&F&6#F4F(-F *6#7%F4\"\"!F.F(-F06#,$F3!\"$F(F(*(&F&6#\"\"$F(-F*6#7%F;F4F.F(F " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 57 "A :=matrix([[17, 20, 40], [-5, -8, -10], [-5 , -5, -13]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'MATRIXG6#7%7 %\"#<\"#?\"#S7%!\"&!\")!#57%F.F.!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{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(6#7%!\"\"F%F3" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 45 "Therefore the general solution is of \+ the form" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "c[1]*exp(5*t)*vector([-1, 1, 0]) + c[2]*exp(2*t)*vec tor([-4, 1, 1]) + c[3]*exp(-t)*vector([-3, 1, 1]);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,(*(&%\"cG6#\"\"\"F(-%$expG6#,$%\"tG\"\"&F(-%'VECTORG 6#7%!\"\"F(\"\"!F(F(*(&F&6#\"\"#F(-F*6#,$F-F8F(-F06#7%!\"%F(F(F(F(*(&F &6#\"\"$F(-F*6#,$F-F3F(-F06#7%!\"$F(F(F(F(" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 88 "augment(vector([-1, 1, 0]), vector([-4, 1, 1]), vec tor([-3, 1, 1]), vector([5, 1, -4]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7&!\"\"!\"%!\"$\"\"&7&\"\"\"F-F-F-7&\"\"!F-F-F)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "rref(\");" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%'MATRIXG6#7%7&\"\"\"\"\"!F)\"\"&7&F)F(F)\"\"#7&F)F) F(!\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 122 "vector([x(t), y( t), z(t)]) = 5*exp(5*t)*vector([-1, 1, 0]) + 2*exp(2*t)*vector([-4, 1, 1]) - 6*exp(-t)*vector([-3, 1, 1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/-%'VECTORG6#7%-%\"xG6#%\"tG-%\"yGF*-%\"zGF*,(*&-%$expG6#,$F+\"\"&\" \"\"-F%6#7%!\"\"F7\"\"!F7F6*&-F36#,$F+\"\"#F7-F%6#7%!\"%F7F7F7FA*&-F36 #,$F+F;F7-F%6#7%!\"$F7F7F7!\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "soln := map( u -> evalm(u) , \" );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%solnG/-%'VECTORG6#7%-%\"xG6#%\"tG-%\"yGF,-%\"zGF,-F' 6#7%,(-%$expG6#,$F-\"\"&!\"&-F76#,$F-\"\"#!\")-F76#,$F-!\"\"\"#=,(F6F: F " 0 "" {MPLTEXT 1 0 116 "soln_list := \{\} :\nfor i from 1 to 3 do\n s[i] := lhs(soln)[i] = rhs(soln)[i];\n s oln_list := \{op(soln_list),\"\}:\nod;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"sG6#\"\"\"/-%\"xG6#%\"tG,(-%$expG6#,$F,\"\"&!\"&-F/6#,$F,\" \"#!\")-F/6#,$F,!\"\"\"#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*soln_l istG<#/-%\"xG6#%\"tG,(-%$expG6#,$F*\"\"&!\"&-F-6#,$F*\"\"#!\")-F-6#,$F *!\"\"\"#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"sG6#\"\"#/-%\"yG6#% \"tG,(-%$expG6#,$F,\"\"&F2-F/6#,$F,F'F'-F/6#,$F,!\"\"!\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*soln_listG<$/-%\"xG6#%\"tG,(-%$expG6#,$F*\" \"&!\"&-F-6#,$F*\"\"#!\")-F-6#,$F*!\"\"\"#=/-%\"yGF),(F,F0F2F5F7!\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"sG6#\"\"$/-%\"zG6#%\"tG,&-%$exp G6#,$F,\"\"#F2-F/6#,$F,!\"\"!\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% *soln_listG<%/-%\"xG6#%\"tG,(-%$expG6#,$F*\"\"&!\"&-F-6#,$F*\"\"#!\")- F-6#,$F*!\"\"\"#=/-%\"yGF),(F,F0F2F5F7!\"'/-%\"zGF),&F2F5F7F@" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "for i from 1 to 3\ndo simpl ify(subs(t = 0, soln_list[i])):\nrhs(\")-rhs(iv[i]);\nod;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#\"\"!\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#\"\"! \"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"zG6#\"\"!!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# \"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "Another way to see that our solution is correct is to compare it with the blackbox solution:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "dsolve(\{ode[1],ode[2],ode[3],iv[1] ,iv[2],iv[3]\},\{x(t), y(t), z(t)\} );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/-%\"xG6#%\"tG,(-%$expG6#,$F(\"\"&!\"&-F+6#,$F(\"\"#!\")-F+6#, $F(!\"\"\"#=/-%\"yGF',(F*F.F0F3F5!\"'/-%\"zGF',&F0F3F5F>" }}}{PARA 4 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 81 "Solving a Firs t Order Linear System Case 2: Defects and Generalized Eigenvectors " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "We will \+ solve the system" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 268 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 19 1 "diff(x[1](t),t) = 13*x[1](t)-8*x[2]( t)-5*x[4](t)+47*x[5](t)" "/-%%diffG6$-&%\"xG6#\"\"\"6#%\"tGF,,**&\"#8F *-&F(6#F*6#F,F*F**&\"\")F*-&F(6#\"\"#6#F,F*!\"\"*&\"\"&F*-&F(6#\"\"%6# F,F*F;*&\"#ZF*-&F(6#F=6#F,F*F*" }}{PARA 269 "" 0 "" {TEXT -1 3 " " } }{PARA 270 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 19 1 "diff(x[2](t),t) = -9*x[1](t)+4*x[2](t)+3*x[4](t)-30*x[5](t)" "/-%%diffG6$-&%\"xG6#\"\"# 6#%\"tGF,,**&\"\"*\"\"\"-&F(6#F06#F,F0!\"\"*&\"\"%F0-&F(6#F*6#F,F0F0*& \"\"$F0-&F(6#F76#F,F0F0*&\"#IF0-&F(6#\"\"&6#F,F0F5" }}{PARA 271 "" 0 " " {TEXT -1 0 "" }}{PARA 272 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 19 1 " diff(x[3](t),t) = 2*x[3](t)+x[4](t)-x[5](t)" "/-%%diffG6$-&%\"xG6#\"\" $6#%\"tGF,,(*&\"\"#\"\"\"-&F(6#F*6#F,F0F0-&F(6#\"\"%6#F,F0-&F(6#\"\"&6 #F,!\"\"" }}{PARA 273 "" 0 "" {TEXT -1 0 "" }}{PARA 274 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 19 1 "diff(x[4](t),t) = -5*x[1](t)+4*x[2](t)+5*x [4](t)-23*x[5](t)" "/-%%diffG6$-&%\"xG6#\"\"%6#%\"tGF,,**&\"\"&\"\"\"- &F(6#F06#F,F0!\"\"*&F*F0-&F(6#\"\"#6#F,F0F0*&F/F0-&F(6#F*6#F,F0F0*&\"# BF0-&F(6#F/6#F,F0F5" }}{PARA 275 "" 0 "" {TEXT -1 0 "" }}{PARA 276 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 19 1 "diff(x[5](t),t) = -6*x[1](t)+4* x[2](t)+2*x[4](t)-22*x[5](t)" "/-%%diffG6$-&%\"xG6#\"\"&6#%\"tGF,,**& \"\"'\"\"\"-&F(6#F06#F,F0!\"\"*&\"\"%F0-&F(6#\"\"#6#F,F0F0*&F;F0-&F(6# F76#F,F0F0*&\"#AF0-&F(6#F*6#F,F0F5" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "The coeff icient Matrix is:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "A := matrix([[13, -8, 0, -5, 47], [-9, 4, 0, 3, -30], [0, 0, 2, 1, -1], [-5, 4, 0, 5, -23], [-6, 4, 0, 2, -22]]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'MATRIXG6#7'7'\"#8!\")\"\" !!\"&\"#Z7'!\"*\"\"%F,\"\"$!#I7'F,F,\"\"#\"\"\"!\"\"7'F-F1F,\"\"&!#B7' !\"'F1F,F5!#A" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "Let us enter the system of equations:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 149 "vector_eqn \+ := map(z -> diff(z,t), vector([x[1](t),x[2](t),x[3](t),x[4](t),x[5](t) ])) =\nevalm(A &* vector([x[1](t),x[2](t),x[3](t),x[4](t),x[5](t)])); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%+vector_eqnG/-%'VECTORG6#7'-%%di ffG6$-&%\"xG6#\"\"\"6#%\"tGF3-F+6$-&F/6#\"\"#F2F3-F+6$-&F/6#\"\"$F2F3- F+6$-&F/6#\"\"%F2F3-F+6$-&F/6#\"\"&F2F3-F'6#7',*F-\"#8F6!\")FB!\"&FH\" #Z,*F-!\"*F6FEFBF?FH!#I,(F " 0 "" {MPLTEXT 1 0 75 "for i from 1 \+ to 5 do\neqn[i] := lhs(vector_eqn)[i] = rhs(vector_eqn)[i];\nod;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$eqnG6#\"\"\"/-%%diffG6$-&%\"xGF&6# %\"tGF0,*F,\"#8-&F.6#\"\"#F/!\")-&F.6#\"\"%F/!\"&-&F.6#\"\"&F/\"#Z" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$eqnG6#\"\"#/-%%diffG6$-&%\"xGF&6#% \"tGF0,*-&F.6#\"\"\"F/!\"*F,\"\"%-&F.6#F7F/\"\"$-&F.6#\"\"&F/!#I" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$eqnG6#\"\"$/-%%diffG6$-&%\"xGF&6#% \"tGF0,(F,\"\"#-&F.6#\"\"%F/\"\"\"-&F.6#\"\"&F/!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$eqnG6#\"\"%/-%%diffG6$-&%\"xGF&6#%\"tGF0,*-&F.6# \"\"\"F/!\"&-&F.6#\"\"#F/F'F,\"\"&-&F.6#F;F/!#B" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$eqnG6#\"\"&/-%%diffG6$-&%\"xGF&6#%\"tGF0,*-&F.6#\" \"\"F/!\"'-&F.6#\"\"#F/\"\"%-&F.6#F;F/F:F,!#A" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "Next the eigenvalues:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "charpoly(A, lambda);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,.*$%'lambdaG\"\"&\"\"\"*$F%\"\"%!\"#* $F%\"\"$!\")*$F%\"\"#\"#;F%F0!#KF'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "factor(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&%' lambdaG\"\"\"\"\"#F&F',&F%F&!\"#F&\"\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "Next we find the eigenvectors that correspond to " }{TEXT 297 1 " " }{XPPEDIT 298 1 "lambda=2" "/%'lambd aG\"\"#" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "evalm(A - 2*array(identity,1..5,1.. 5));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7'7'\"#6!\")\"\"!! \"&\"#Z7'!\"*\"\"#F*\"\"$!#I7'F*F*F*\"\"\"!\"\"7'F+\"\"%F*F0!#B7'!\"'F 6F*F/!#C" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "mulrow(\", 1, 1 /11); # multiply row 1 by 1/11" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'M ATRIXG6#7'7'\"\"\"#!\")\"#6\"\"!#!\"&F+#\"#ZF+7'!\"*\"\"#F,\"\"$!#I7'F ,F,F,F(!\"\"7'F.\"\"%F,F4!#B7'!\"'F9F,F3!#C" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "addrow(\", 1, 2, 9); # add 9 times row 1 to row 2 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7'7'\"\"\"#!\")\"#6\" \"!#!\"&F+#\"#ZF+7'F,#!#]F+F,#!#7F+#\"#$*F+7'F,F,F,F(!\"\"7'F.\"\"%F, \"\"$!#B7'!\"'F;F,\"\"#!#C" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "addrow(\", 1, 4, 5); # add 5 times row 1 to row 4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7'7'\"\"\"#!\")\"#6\"\"!#!\"&F+#\"#ZF+ 7'F,#!#]F+F,#!#7F+#\"#$*F+7'F,F,F,F(!\"\"7'F,#\"\"%F+F,#\"\")F+#!#=F+7 '!\"'F " 0 "" {MPLTEXT 1 0 48 "addrow (\", 1, 5, 6); # add 6 times row 1 to row 5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7'7'\"\"\"#!\")\"#6\"\"!#!\"&F+#\"#ZF+7'F,# !#]F+F,#!#7F+#\"#$*F+7'F,F,F,F(!\"\"7'F,#\"\"%F+F,#\"\")F+#!#=F+7'F,#! \"%F+F,F)#\"#=F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "addrow( \", 4, 5, 1); # add row 4 to row 5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# -%'MATRIXG6#7'7'\"\"\"#!\")\"#6\"\"!#!\"&F+#\"#ZF+7'F,#!#]F+F,#!#7F+# \"#$*F+7'F,F,F,F(!\"\"7'F,#\"\"%F+F,#\"\")F+#!#=F+7'F,F,F,F,F," }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "mulrow(\", 2, -11/50); # mul tiply row 2 by -11/50" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7 '7'\"\"\"#!\")\"#6\"\"!#!\"&F+#\"#ZF+7'F,F(F,#\"\"'\"#D#!#$*\"#]7'F,F, F,F(!\"\"7'F,#\"\"%F+F,#\"\")F+#!#=F+7'F,F,F,F,F," }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 54 "addrow(\", 2, 1, 8/11); # add 8/11 times row 2 to row 1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7'7'\"\"\" \"\"!F)#!\"(\"#D#\"#tF,7'F)F(F)#\"\"'F,#!#$*\"#]7'F)F)F)F(!\"\"7'F)#\" \"%\"#6F)#\"\")F:#!#=F:7'F)F)F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "addrow(\", 2, 4, -4/11); # add -4/11 times row 2 to r ow 4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7'7'\"\"\"\"\"!F)# !\"(\"#D#\"#tF,7'F)F(F)#\"\"'F,#!#$*\"#]7'F)F)F)F(!\"\"7'F)F)F)#\"#;F, #!#CF,7'F)F)F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "addro w(\", 3, 1, 7/25); # add 7/25 times row 3 to row 1" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%'MATRIXG6#7'7'\"\"\"\"\"!F)F)#\"#m\"#D7'F)F(F)#\"\" 'F,#!#$*\"#]7'F)F)F)F(!\"\"7'F)F)F)#\"#;F,#!#CF,7'F)F)F)F)F)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "addrow(\", 3, 2, -6/25); # a dd -6/25 times row 3 to row 2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MA TRIXG6#7'7'\"\"\"\"\"!F)F)#\"#m\"#D7'F)F(F)F)#!#\")\"#]7'F)F)F)F(!\"\" 7'F)F)F)#\"#;F,#!#CF,7'F)F)F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "addrow(\", 3, 4, -16/25); # add -16/25 times row 3 to row 4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7'7'\"\"\"\"\"!F )F)#\"#m\"#D7'F)F(F)F)#!#\")\"#]7'F)F)F)F(!\"\"7'F)F)F)F)#!\")F,7'F)F) F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "mulrow(\", 4, -25 /8); # multiply row 4 by -25/8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'M ATRIXG6#7'7'\"\"\"\"\"!F)F)#\"#m\"#D7'F)F(F)F)#!#\")\"#]7'F)F)F)F(!\" \"7'F)F)F)F)F(7'F)F)F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "addrow(\", 4, 1, -66/25); # add -66/25 times row 4 to row 1" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7'7'\"\"\"\"\"!F)F)F)7'F)F (F)F)#!#\")\"#]7'F)F)F)F(!\"\"7'F)F)F)F)F(7'F)F)F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "addrow(\", 4, 2, 81/50); # add 81/5 0 times row 4 to row 2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6# 7'7'\"\"\"\"\"!F)F)F)7'F)F(F)F)F)7'F)F)F)F(!\"\"7'F)F)F)F)F(7'F)F)F)F) F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "addrow(\", 4, 3, 1); \+ # add row 4 to row 3 *** FINISHED ***" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7'7'\"\"\"\"\"!F)F)F)7'F)F(F)F)F)7'F)F)F)F(F)7'F)F)F)F) F(7'F)F)F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "rref(A - \+ 2*array(identity,1..5,1..5)); # A one-step way to get to the same plac e" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7'7'\"\"\"\"\"!F)F)F) 7'F)F(F)F)F)7'F)F)F)F(F)7'F)F)F)F)F(7'F)F)F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "Every eigenvector is a multiple of " }{TEXT 299 15 "[0, 0, 1, 0, 0]" }{TEXT -1 80 ". There is a defect of (3 - 1) or 2. We must fi nd two generalized eigenvectors." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "evalm( (A - 2*array(identity ,1..5,1..5))^3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7'7'\" $c#!$%Q\"\"!F*\"%;77'!$#>\"$C#F*F*!$;)7'F*F*F*F*F*7'!$G\"\"$#>F*F*!$3' F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "rref(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7'7'\"\"\"\"\"!F)F)#\"\"&\"\"#7'F)F (F)F)#!\"$F,7'F)F)F)F)F)F0F0" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 18 "This tells us that" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "u[3] := vect or([5, -3, 0, 0, -2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"uG6#\" \"$-%'VECTORG6#7'\"\"&!\"$\"\"!F.!\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 30 "is a generalized eigenvector. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Check:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "evalm( (A - 2*array(identity,1..5,1..5))^3 &* u[3]);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%'VECTORG6#7'\"\"!F'F'F'F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "Now, the next in the chai n of generalized eigenvectors:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "u[2] := evalm( (A - 2*array( identity,1..5,1..5)) &* u[3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&% \"uG6#\"\"#-%'VECTORG6#7'!#:\"\"*F'F-\"\"'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "and finally the eigenvector:" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "u[1] := evalm( (A - 2*array(identity,1..5,1..5)) &* u[2]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"uG6#\"\"\"-%'VECTORG6#7'\"\"!F,\" \"$F,F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "and the part of the general solution that corresponds to " } {TEXT 300 1 " " }{XPPEDIT 301 1 "lambda=2" "/%'lambdaG\"\"#" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "part1 := c[1]*u[1]*exp (2*t) + c[2]*(u[1]*t + u[2])*exp(2*t) + c[3]*(u[1]*t^2/2 + u[2]*t + u[ 3])*exp(2*t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&part1G,(*(&%\"cG6# \"\"\"F*&%\"uGF)F*-%$expG6#,$%\"tG\"\"#F*F**(&F(6#F2F*,&*&F+F*F1F*F*&F ,F5F*F*F-F*F**(&F(6#\"\"$F*,(*&F+F*F1F2#F*F2*&F8F*F1F*F*&F,F;F*F*F-F*F *" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 14 "Verification:\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "for i from 1 to 5 do\nsoln[i] := x[i](t) = evalm(p art1)[i];\nod:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "check := [] :\nfor i from 1 to 5 do\nsimplify(subs(\{soln[1],soln[2],soln[3],soln[ 4],soln[5]\},eqn[i])):\nlhs(\")-rhs(\"):\ncheck := [op(check),\"]:\nod :" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "check;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'\"\"!F$F$F $F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 49 "Next we find the eigenvectors that corres pond to " }{TEXT 302 1 " " }{XPPEDIT 303 1 "lambda=-2" "/%'lambdaG,$\" \"#!\"\"" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "evalm( A - (-2)*array(identity,1..5 , 1..5) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7'7'\"#:!\") \"\"!!\"&\"#Z7'!\"*\"\"'F*\"\"$!#I7'F*F*\"\"%\"\"\"!\"\"7'F+F3F*\"\"(! #B7'!\"'F3F*\"\"#!#?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "rref (\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7'7'\"\"\"\"\"!F) F)\"\"#7'F)F(F)F)#!\"$F*7'F)F)F(F)F)7'F)F)F)F(!\"\"7'F)F)F)F)F)" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "Every eig envector is a multiple of " }{TEXT 304 19 "[4, -3, 0, -2, -2]" } {TEXT -1 42 ". There is a defect of (2-1) or 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "evalm(( A - (-2)*array(identity,1..5,1..5))^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7'7'\"#S\"\"!F)!#S\"$?\"7'!#CF)F)\"#C!#s7'\"\"\"F)\" #;\"\"*!\"(7'!\")F)F)F(!#c7'!#;F)F)F2!#[" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "rref(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIX G6#7'7'\"\"\"\"\"!F)F)\"\"#7'F)F)F(F)F)7'F)F)F)F(!\"\"7'F)F)F)F)F)F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Choose " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "v[2] := vector([2, 1, 0, -1, -1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"vG6#\"\"#-%'VECTORG6#7' F'\"\"\"\"\"!!\"\"F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "Set" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "v[1] := eval m((A - (-2)*array(identity,1..5,1..5)) &* v[2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"vG6#\"\"\"-%'VECTORG6#7'!#?\"#:\"\"!\"#5F/" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "The part of the general solution that correspon ds to " }{TEXT 307 1 " " }{XPPEDIT 308 1 "lambda=-2" "/%'lambdaG,$\" \"#!\"\"" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "part2 := c[4]*v[1]*exp(-2*t) + c[5]*(v[1]*t + v[2])*exp(-2*t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&part2G,&*(&%\"cG6#\"\"%\"\"\"&%\"vG6#F+F+-%$ expG6#,$%\"tG!\"#F+F+*(&F(6#\"\"&F+,&*&F,F+F3F+F+&F-6#\"\"#F+F+F/F+F+ " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "Veri fication:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "for i from 1 to 5 do\nsoln[i] := x[i](t) = evalm(part 2)[i];\nod:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "check := []:\nfor i from 1 to 5 do\nsimplify(subs(\{soln[1],soln[2],soln[3],soln[4],sol n[5]\},eqn[i])):\nlhs(\")-rhs(\"):\ncheck := [op(check),\"]:\nod:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "check;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'\"\"!F$F$F$F$" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "The full general sol ution:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "part1 + part2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,* (&%\"cG6#\"\"\"F(&%\"uGF'F(-%$expG6#,$%\"tG\"\"#F(F(*(&F&6#F0F(,&*&F)F (F/F(F(&F*F3F(F(F+F(F(*(&F&6#\"\"$F(,(*&F)F(F/F0#F(F0*&F6F(F/F(F(&F*F9 F(F(F+F(F(*(&F&6#\"\"%F(&%\"vGF'F(-F,6#,$F/!\"#F(F(*(&F&6#\"\"&F(,&*&F DF(F/F(F(&FEF3F(F(FFF(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "Verification:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "for i from 1 to 5 do\nsoln[i] := x[ i](t) = evalm(part1)[i]+evalm(part2)[i];\nod:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "check := []:\nfor i from 1 to 5 do\nsimplify(subs (\{soln[1],soln[2],soln[3],soln[4],soln[5]\},eqn[i])):\nlhs(\")-rhs(\" ):\ncheck := [op(check),\"]:\nod:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "map(simplify, check);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'\"\"!F$F$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "A fundamental matrix is obtained as follows:" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 154 "Phi \+ := tau -> subs(t = tau, augment(u[1]*exp(2*t), (u[1]*t+u[2])*exp(2*t), (1/2*u[1]*t^2+u[2]*t+u[3])*exp(2*t), v[1]*exp(-2*t), (v[1]*t+v[2])*ex p(-2*t))):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "Phi(t);\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7'7'\"\"!,$-%$expG6#,$%\"t G\"\"#!#:*&F*\"\"\",&F.F0\"\"&F2F2,$-F+6#,$F.!\"#!#?*&F6F2,&F.F:F/F2F2 7'F(,$F*\"\"**&F*F2,&F.F?!\"$F2F2,$F6\"#:*&F6F2,&F.FDF2F2F27',$F*\"\"$ *&F*F2,&F.FIF/F2F2*&F*F2,&*$F.F/#FIF/F.F/F2F(F(7'F(F>,$*&F*F2F.F2F?,$F 6\"#5*&F6F2,&F.FT!\"\"F2F27'F(,$F*\"\"'*&F*F2,&F.FZF9F2F2FSFU" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 23 "Exponential of a Matrix" }}{PARA 0 "" 0 " " {TEXT -1 38 "To find the exponential of a matrix " }{TEXT 309 2 "t A" }{TEXT -1 61 " you can solve a system, finding the fundamental ma trix " }{XPPEDIT 312 1 "Phi(t)" "-%$PhiG6#%\"tG" }{TEXT -1 34 ". T hen form the matrix product " }{XPPEDIT 310 1 "Phi(t)*Phi(0)^(-1)" " *&-%$PhiG6#%\"tG\"\"\")-F$6#\"\"!,$F'!\"\"F'" }{TEXT 311 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "For examp le," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "exp1 := map(simplify, evalm(Phi(t) &* inverse(Phi(0)) ));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%exp1G-%'MATRIXG6#7'7',$*&,(- %$expG6#,$%\"tG\"\"%!\"&*&F-\"\"\"F1F5\"\"&F2F5F5-F.6#,$F1!\"#F5!\"\", $*&F7F5F1F5!\")\"\"!,$*&-F.6#,$F1\"\"#F5F1F5F3,$*&,*F-!#5F4F6\"#5F5F1! #7F5F7F5F;7',$*&,(F-F;F4F5F5F5F5F7F5\"\"$,&F7F5F=\"\"'F?,$FAFP,$*&,*F- F:F4F5FEF5F1!\"$F5F7F5FP7',$*&FBF5F1FE#F5FEF?FB,&FAF5FZFen,&FAF;FZFen7 '*&,(F-F:F4FPFEF5F5F7F5,$F=F2F?,&FBF5FAFP*&,*F-F3F4FPF6F5F1!\"'F5F7F57 ',$FNFEF[oF?,$FAFE*&,*F-!\"%F4FEF6F5F1F_oF5F7F5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "There is also a formula. If " }{TEXT 313 1 " " }{XPPEDIT 314 1 "\{lambda[1],lambda[2],`...`,lambda[k]\}" "<&&%'lambdaG6#\"\"\"& F$6#\"\"#%$...G&F$6#%\"kG" }{TEXT -1 36 " is the set of eigenvalues of " }{TEXT 315 1 "A" }{TEXT -1 10 " with " }{XPPEDIT 317 1 "m [i]" "&%\"mG6#%\"iG" }{TEXT -1 24 " the multiplicity of " } {XPPEDIT 316 1 "lambda[i]" "&%'lambdaG6#%\"iG" }{TEXT -1 10 " , if \+ " }{TEXT 318 1 " " }{XPPEDIT 319 1 "p(lambda)=(lambda-lambda[1])^m[1] *(lambda-lambda[2])^m[2]*`...`*(lambda-lambda[k])^m[k]" "/-%\"pG6#%'la mbdaG**),&F&\"\"\"&F&6#F*!\"\"&%\"mG6#F*F*),&F&F*&F&6#\"\"#F-&F/6#F5F* %$...GF*),&F&F*&F&6#%\"kGF-&F/6#F=F*" }{TEXT -1 13 ", if " } {TEXT 320 1 " " }{XPPEDIT 321 1 "q[i](lambda)=p(lambda)/(lambda-lambda [i])^m[i]" "/-&%\"qG6#%\"iG6#%'lambdaG*&-%\"pG6#F)\"\"\"),&F)F.&F)6#F' !\"\"&%\"mG6#F'F3" }{TEXT -1 13 ", if " }{TEXT 322 1 " " } {XPPEDIT 323 1 "\{a[1](lambda),a[2](lambda),`...`,a[k](lambda)\}" "<&- &%\"aG6#\"\"\"6#%'lambdaG-&F%6#\"\"#6#F)%$...G-&F%6#%\"kG6#F)" }{TEXT -1 64 ", is the family of uniquely determined polynomials such that \+ " }{XPPEDIT 324 1 "deg(a[i])<=m[i]-1" "1-%$degG6#&%\"aG6#%\"iG,&&%\"m G6#F)\"\"\"F.!\"\"" }{TEXT -1 19 ", and \n\n " }{XPPEDIT 325 1 "1=a[1](lambda)*q[1](lambda)+a[2](lambda)*q[2](lambda)+`...`+a[k ](lambda)*q[k](lambda)" "/\"\"\",**&-&%\"aG6#F#6#%'lambdaGF#-&%\"qG6#F #6#F+F#F#*&-&F(6#\"\"#6#F+F#-&F.6#F56#F+F#F#%$...GF#*&-&F(6#%\"kG6#F+F #-&F.6#F@6#F+F#F#" }{TEXT -1 8 ",\n\n then" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 " " } {TEXT 326 11 " " }{XPPEDIT 327 1 "exp(t*A) = Sum(exp(lambda[ i]*t)*a[i](A)*q[i](A)*Sum( (A-lambda[i]*I)^(` `^j)*t^j/j!,j=0..m[i]- 1),i=1..k)" "/-%$expG6#*&%\"tG\"\"\"%\"AGF(-%$SumG6$**-F$6#*&&%'lambda G6#%\"iGF(F'F(F(-&%\"aG6#F46#F)F(-&%\"qG6#F46#F)F(-F+6$*(),&F)F(*&&F26 #F4F(%\"IGF(!\"\")%\"~G%\"jGF()F'FKF(-%*factorialG6#FKFH/FK;\"\"!,&&% \"mG6#F4F(F(FHF(/F4;F(%\"kG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 328 18 "Computational \+ Tip:" }}{PARA 0 "" 0 "" {TEXT -1 11 "The set " }{TEXT 329 1 " " } {XPPEDIT 330 1 "\{a[1](lambda),a[2](lambda),`...`,a[k](lambda)\}" "<&- &%\"aG6#\"\"\"6#%'lambdaG-&F%6#\"\"#6#F)%$...G-&F%6#%\"kG6#F)" }{TEXT -1 69 " is found by calculating the partial fraction decompositio n of " }{TEXT 331 2 " " }{XPPEDIT 332 1 "1/p(lambda)" "*&\"\"\"F#-% \"pG6#%'lambdaG!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Example:" }}{PARA 0 "" 0 "" {TEXT -1 41 "For the matrix that we have been using," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "p(lambda) = (lambd a+2)^2*(lambda-2)^3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"pG6#%'lam bdaG*&,&F'\"\"\"\"\"#F*F+,&F'F*!\"#F*\"\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "partial_fraction_decomp := 1/((lambda+2)^2*(lambda-2)^3)= c[1]/(lambda+2)+c[2]/(lambda+2)^2+c [3]/(lambda-2)+c[4]/(lambda-2)^2+c[5]/(lambda-2)^3;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%8partial_fraction_decompG/*&,&%'lambdaG\"\"\"\"\"#F )!\"#,&F(F)F+F)!\"$,,*&&%\"cG6#F)F)F'!\"\"F)*&&F16#F*F)F'F+F)*&&F16#\" \"$F)F,F3F)*&&F16#\"\"%F)F,F+F)*&&F16#\"\"&F)F,F-F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "solve(identity(\",lambda),\{c[1],c[2],c[3 ],c[4],c[5]\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<'/&%\"cG6#\"\"%#! \"\"\"#K/&F&6#\"\"$#F/\"$c#/&F&6#\"\"&#\"\"\"\"#;/&F&6#F7#!\"$F1/&F&6# \"\"##F*\"#k" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "subs(\", pa rtial_fraction_decomp);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&,&%'lamb daG\"\"\"\"\"#F'!\"#,&F&F'F)F'!\"$,,*$F%!\"\"#F+\"$c#*$F%F)#F.\"#k*$F* F.#\"\"$F0*$F*F)#F.\"#K*$F*F+#F'\"#;" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 7 "We get:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "a[1] := lambda -> -3/25 6*(lambda+2) - 1/64;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#\"\" \":6#%'lambdaG6\"6$%)operatorG%&arrowGF+,&9$#!\"$\"$c##!\"&\"$G\"F'F+F +" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "a[2] := lambda -> 3/25 6*(lambda-2)^2 -(1/32)*(lambda-2)+1/16;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#\"\"#:6#%'lambdaG6\"6$%)operatorG%&arrowGF+,(*$,&9$\"\" \"!\"#F3F'#\"\"$\"$c#F2#!\"\"\"#K#F3\"\")F3F+F+" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 31 "q[1] := lambda -> (lambda-2)^3;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>&%\"qG6#\"\"\":6#%'lambdaG6\"6$%)operatorG%&arr owGF+*$,&9$F'!\"#F'\"\"$F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "q[2] := lambda -> (lambda+2)^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"qG6#\"\"#:6#%'lambdaG6\"6$%)operatorG%&arrowGF+*$,&9$\"\"\"F'F 2F'F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 7 "Check:\n" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 66 "testeq(1 = a[1](lambda)*q[1](lambda) + a[2](la mbda)*q[2](lambda));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "lambda[1] := -2; lambda[2] \+ := 2; m[1] := 2; m[2] := 3; k := 2; i := 'i': j := 'j':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'lambdaG6#\"\"\"!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'lambdaG6#\"\"#F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>&%\"mG6#\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"mG6#\"\" #\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"kG\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Id := array(identity,1..5,1..5);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#IdG-%&arrayG6&%)identityG;\"\"\"\" \"&F)7\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "term1 := subs(i =1,exp(lambda[i]*t)*a[i](A)*q[i](A)\n *(Id + (A-lambda [i]*Id)*t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&term1G**-%$expG6#*& &%'lambdaG6#\"\"\"F-%\"tGF-F--&%\"aGF,6#%\"AGF--&%\"qGF,F2F-,&%#IdGF-* &,&F3F-*&F*F-F8F-!\"\"F-F.F-F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalm(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRI XG6#7'7',$-%$expG6#,$%\"tG!\"#!\"%,$*&F)\"\"\"F-F2!\")\"\"!F4*&F)F2,&F -\"#7!#5F2F27',$F)\"\"$*&F)F2,&F-\"\"'F2F2F2F4F4*&F)F2,&F-!\"*F>F2F27' F4F4F4F4F47',$F)\"\"#,$F1\"\"%F4F4*&F)F2,&F-!\"'\"\"&F2F2FC" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "term2 := subs(i=2,exp(lambd a[i]*t)*a[i](A)*q[i](A)\n *(Id + (A-lambda[i]*Id)*t + \+ (A-lambda[i]*Id)^2*t^2/2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&term 2G**-%$expG6#*&&%'lambdaG6#\"\"#\"\"\"%\"tGF.F.-&%\"aGF,6#%\"AGF.-&%\" qGF,F3F.,(%#IdGF.*&,&F4F.*&F*F.F9F.!\"\"F.F/F.F.*&F;F-F/F-#F.F-F." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalm(\");" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%'MATRIXG6#7'7'*&-%$expG6#,$%\"tG\"\"#\"\"\",&\"\"&F /F-!\"&F/\"\"!F3,$*&F)F/F-F/F2*&F)F/,&F-F2\"#5F/F/7'*&F)F/,&!\"$F/F-\" \"$F/F3F3,$F5F=*&F)F/,&F-F=!\"'F/F/7',$*&F)F/F-F.#F/F.F3F)*&F)F/,&F-F/ *$F-F.FEF/*&F)F/,&F-!\"\"FHFEF/7'*&F)F/,&!\"#F/F-F=F/F3F3*&F)F/,&F/F/F -F=F/*&F)F/,&F-F=F2F/F/7'*&F)F/,&FOF/F-F.F/F3F3,$F5F.*&F)F/,&F-F.!\"%F /F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "term1 + term2;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&**-%$expG6#,$%\"tG!\"#\"\"\",&%\"AG# !\"$\"$c##!\"&\"$G\"F+F+,&F-F+F*F+\"\"$,&%#IdGF+*&,&F-F+F7\"\"#F+F)F+F +F+F+**-F&6#,$F)F:F+,(*$F4F:#F5F0F-#!\"\"\"#K#F+\"\")F+F+,&F-F+F:F+F:, (F7F+*&,&F-F+F7F*F+F)F+F+*&FJF:F)F:#F+F:F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "exp2 := evalm(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%exp2G-%'MATRIXG6#7'7',$*&,(-%$expG6#,$%\"tG\"\"%!\"&*&F-\"\" \"F1F5\"\"&F2F5F5-F.6#,$F1!\"#F5!\"\",$*&F7F5F1F5!\")\"\"!,$*&-F.6#,$F 1\"\"#F5F1F5F3,$*&,*F-!#5F4F6\"#5F5F1!#7F5F7F5F;7',$*&,(F-F;F4F5F5F5F5 F7F5\"\"$,&F7F5F=\"\"'F?,$FAFP,$*&,*F-F:F4F5FEF5F1!\"$F5F7F5FP7',$*&FB F5F1FE#F5FEF?FB,&FAF5FZFen,&FAF;FZFen7'*&,(F-F:F4FPFEF5F5F7F5,$F=F2F?, &FBF5FAFP*&,*F-F3F4FPF6F5F1!\"'F5F7F57',$FNFEF[oF?,$FAFE*&,*F-!\"%F4FE F6F5F1F_oF5F7F5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "Let us check this with the expression obtained by using t he fundamental matrix:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "map(sim plify, evalm(exp1-exp2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIX G6#7'7'\"\"!F(F(F(F(F'F'F'F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 33 "There is also a one-step command:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "exponential(t*A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7 '7',(-%$expG6#,$%\"tG\"\"#\"\"&*&F)\"\"\"F-F1!\"&-F*6#,$F-!\"#!\"%,$*& F3F1F-F1!\")\"\"!,$F0F2,*F9\"#7F3!#5F)\"#5F0F27',(F0\"\"$F3FCF)!\"$,&F 3F1F9\"\"'F;,$F0FC,*F9!\"*F3FFF)!\"'F0FC7',$*&F)F1F-F.#F1F.F;F),&F0F1F MFN,&F0!\"\"FMFN7',(F0FCF3F.F)F6,$F9\"\"%F;,&F)F1F0FC,*F9FJF3F/F)F2F0F C7',(F0F.F3F.F)F6FTF;,$F0F.,*F)F7F9FJF3F/F0F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "Let us check this:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "map(simplify, evalm(\" - exp 1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7'7'\"\"!F(F(F(F(F 'F'F'F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 32 "Copyright and Author Information" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 48 "5.1-5.8epR4.mws A M aple Release 4 worksheet." }}{PARA 261 "" 0 "" {TEXT -1 0 "" }}{PARA 262 "" 0 "" {TEXT -1 24 "Author: Brian E. Blank " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 263 "" 0 "" {TEXT -1 23 "Date: 11 November 2000 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "This \+ document may not be distributed by any medium," }}{PARA 0 "" 0 "" {TEXT -1 55 "including print, disk, and electronic transfer, without" }}{PARA 0 "" 0 "" {TEXT -1 39 "prior written permission of the author. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 264 "" 0 "" {TEXT -1 49 "For more information, please contact the author:" }}{PARA 265 "" 0 "" {TEXT -1 4 " " }}{PARA 265 "" 0 "" {TEXT -1 32 " Department of \+ Mathematics, " }}{PARA 0 "" 0 "" {TEXT -1 39 " Washington Universi ty in St. Louis" }}{PARA 0 "" 0 "" {TEXT -1 26 " St. Louis, MO 6 3130" }}{PARA 0 "" 0 "" {TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 33 " Telephone: (314) 935-6763" }}{PARA 266 "" 0 "" {TEXT -1 44 " e-mail: brian@math.wustl.edu" }}{PARA 267 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "Copyright: \251 2000 Bri an E. Blank, All Rights Reserved." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} }}{MARK "10" 0 }{VIEWOPTS 1 1 0 3 4 1802 }