{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 56 0 0 0 2 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 29 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 2 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 "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 179 0 4 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 "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 2 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 "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 74 0 29 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 "" 0 260 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 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "Matrix Algebra worked exam ple 5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Consider the system" }} {PARA 256 "" 0 "" {OLE 1 3584 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:j::>:B>N: F:nyyyyy]::yyyyyy::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::fyyyyya:nYf::wyyyqy;:::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::NDYmq^H;C:ELq^H_mvJ::: :::::gjR<:T><::;Xr[TM;t_wW:A:;B;B:F:YLpfF>:::::::::J?N Z;vyyyyyY:vYxY:B:::::::c:;:=:jR>@Wlj^HMMufF;J:::::::N=?:xI:;Z::::::j:> :C=;:=j[vGUMrvC?MoJ::::::::JCN:yyyxI:;Z::::::j;B:s<;:wA?Z::C:J=j=B:K:M :OJ:V;;J@>:UJ:n;v;;Jyky;:::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::j:b:klN\\=@i:FZlVjsoZ<:Zs?:R<:TNC>:UTRcETcTX[US;SK]UW=EWMuUWm>j\\@A:::cYH_WV>Z::::::::jysyA :CB:F:;jysy?:;:gI;vo>JmDZ:F:MZ=V`;B:?jDJ:f:;j>D:;B:CZ:NZ;F:E:=b:yyyyI: E:M::Gc;YJCvYxY:JZyyayYZLjALj? JMJ@fc[;>:;I:EZ:F;NZ:vCS=[LsfFaMR>@>Z::::::::kJ;@:;J;>Z:vYxY:>Z::::::j D_=a=[;;B:::::::JF>:yay=J:B::::::nYyA<::::::::::::jysy:>: <:::::::::::::::::::vYxI:;Z:::::::::]:qi:;fy>Z:JBAZ:b:DJ:JSfj=nH=:G;eZ:VY;><:[V:=:;b:^dcgg_WhZ nc_whZNdigg[oGuMIKG:M:_;Ku:=:k=JSFl=>E=B:FIK?N @>j=FZ:jvJB:C:[q:F;;JSdjb^e;F:;B:uKfG;N @]Nn@j:jXjDjw;<:[V:B:D:c\\_;:^?CB:f?=JZ:B:;H:M:djBol< F:B:N;:[Z:VY[j=J:^qX=j; :::::::::3:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "find a) the equili brium point" }}{PARA 0 "" 0 "" {TEXT -1 96 " b) the eigenvalues \015\n c) the eigenvectors\015\n d) the equations of the dire ctrices\015\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "To find the equilibrium point" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "X:=2*x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"XG,$%\" xG\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "Y:=2*y;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"YG ,$%\"yG\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "The equilibrium p oint is obviously (0,0)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "To find the eigenvalues" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "A:=matrix(2,2,[2,0,0,2]);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'MATRIXG6#7$7$\"\"#\"\"!7$F+F *" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "eigenvalues:=eigenvals (A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,eigenvaluesG6$\"\"#F&" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Thus the eigenvalue is 2 twice. " }}{PARA 0 "" 0 "" {TEXT -1 117 "Since the matrix is diagonal there is \+ an infinite number of eigenvectors and hence an infinite number of dir ectrices." }}{PARA 0 "" 0 "" {TEXT -1 24 "To find the eigenvectors" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "eigensystem:=eigenvectors(A );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,eigensystemG7%\"\"#F&<$-%'VEC TORG6#7$\"\"!\"\"\"-F)6#7$F-F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 131 "Taken at face value this imlis that there are only two independen t eigenvectors.That this is incorrect may be shown as follows. Let" }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{OLE 1 3584 1 "[xm]Br=WfoRrB:::wk;ny yI;G:;:j::>:B>N:F:nyyyyy]::yyyyyy::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::fyyyyya:nYf::wyyyqy;:::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::NDYmq^ H;C:ELq^H_mvJ::::::::gjR<:T><::;Xr[TM;t_wW:A:;B;B:F:YL pfF>:::::::::J?NZ;vyyyyyY:vYxY:B:::::::c:;:=:jR>@Wlj^HMMufF;J:::::::N= ?:xI:;Z::::::j:>:C>;:=j[vGUMrvC?MoJ::::::::JCN:yyyxI:;Z::::::j;B:s<;:w A?Z:F;^:nYn:v:J?>:wAQ:SJ:f;;JAjA>:wAyA::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::=Zd:Xb[og]gffk=v:;J:::::j::;P:<:TNC>:UTRcETcTX[US; SK]UW=EWMuUWm>j\\@A:::cYH_WV;Z::::::::jysyA:CB:F:;jysy?:;:Wt:F]?JTD:=j >r:oR:N:eZ:>:EJ:F[n>r:WJ:n^@v;_jysy;B:K:;xyAwy;h:AD;Q:G;Sj`@@J:>XEZ:F;NZ:vCS=[LsfFa MR>@>Z::::::::kJ;@:;J;>Z:vYxY:>Z::::::jD_=a=[;;B:::::::JF >:yay=J:B::::::nYyA<::::::::::::jysy:>:<:::::::::::::::::::vYxI:;Z:::: :::::eB:JSfu=^T=:A=eJ:VY;syB:>l;>:X=j>>:_c<;X:;l:B=:UD:[n>B:_KZG]< F:;ja>l:F;<:sg :>Z:FZX;j;:::::::1:" }}{PARA 0 "" 0 "" {TEXT -1 19 "be any vector. 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