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::::3:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "find  a)  the equilibri
um point" }}{PARA 0 "" 0 "" {TEXT -1 27 "       b)   the eigenvalues" 
}}{PARA 0 "" 0 "" {TEXT -1 70 "       c) the eigenvectors\n       d) t
he equations of the directrices\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "Restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 
"with(linalg):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Enter the equat
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{MPLTEXT 1 0 14 "E1:=4*x+2*y-2;" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "E2:=-x+y+2;" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT 256 30 "To find the equilibrium points" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "solve(\{E1=0,E2=0\},\{x,y\})
;" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 31 "The equilibrium point is (1,-1)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 257 23 "To find the \+
eigenvalues" }{TEXT -1 154 " you first need to enter the matrix A.The \+
first two numbers define a 2 by 2 matrix. The four numbers in the squa
re bracket are the elements of the matrix." }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 26 "A:=matrix(2,2,[4,2,-1,1]);" }}{PARA 11 "" 1 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Then use the 'eig
envals' command" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "eigenval
ues=eigenvals(A);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "there are tw
o real eigenvalues 2 and 3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT 258 25 "To find the eigenvectors " }{TEXT -1 30 
"use the 'eigenvectors' command" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 28 "eigensystem=eigenvectors(A);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 173 "The first number 2 is an eigenvalue and the following 1 tells \+
you that it only occurs once, i.e. it is not repeated. The term [-1,1]
 is the eigenvector for the eigenvalue 2." }}{PARA 0 "" 0 "" {TEXT -1 
85 "The number 3 is the second eigenvalue followed by a 1 which means \+
it is not repeated." }}{PARA 0 "" 0 "" {TEXT -1 51 "The term [-2,1] is
 the eigenvector for the value 3." }}{PARA 0 "" 0 "" {TEXT -1 26 "Thus
 the eigenvectors are " }}{PARA 257 "" 0 "" {OLE 1 4102 1 "[xm]Br=WfoR
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:[q:VZ:B:;::::::::::5:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "find a)
 the equilibrium point" }}{PARA 0 "" 0 "" {TEXT -1 93 "      b)  the e
igenvalues\n      c) the eigenvectors\n     d) the equations of the di
rectrices\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 263 "" 0 "" {TEXT 
-1 29 "To find the equilibrium point" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 25 "First enter the equations" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "E1:=2*x;
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "E2:=2*y;" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "The eq
uilibrium point is obviously (0,0)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 264 "" 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 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 26 "eigenvalues:=eigenvals(A);" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 32 "Thus the eigenvalue is 2 twice. " }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 24 "To find the eigenvect
ors" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "eigensystem:=eigenve
ctors(A);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 133 "Taken at face value this implies that there are only two
 independent eigenvectors.That this is incorrect may be shown as follo
ws. Let" }}{PARA 260 "" 0 "" {TEXT -1 1 " " }{OLE 1 3594 1 "[xm]Br=Wfo
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::::::::::::::::::::::::::::::::::::::::::::::::::::::j:b:<Z::::::::::
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[:>Z:N`Dfm;nV=J:<J:::::::::::jk<JNG;N@ux:cp:jkjD>:qQJxI<:[V:;Z:b::::uM
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"" 0 "" {TEXT -1 19 "be any vector. Then" }}{PARA 261 "" 0 "" {OLE 1 
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:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::fyyyyya
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::5:" }}{PARA 0 "" 0 "" {TEXT -1 17 "Hence the vector " }}{PARA 262 "
" 0 "" {OLE 1 3598 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:j::>:B>N:F:nyyyyy]::
yyyyyy::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
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:::::::::::::::::::::::::::::::::::::NDYmq^H;C:ELq^H_mvJ::::::::gj<vyy
uyJ<j?V:;J:>R<:T><::[eDSBpoIewW:A:;B;B:F:YLpfF>:::::::::J?NZ;vyyyyyY:v
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MrvC?MoJ::::::::JCN:yyyxI:;Z::::::j;B:s<;:wA?Z:F;^:nYn:v:;B:K:OJ:nYV;^
;f;;JAjA>:wAyA::::::::::::::::::::::::::::::::::::::::::::::::::::::::
::::::::::::::::::=Z<B:<::::::::::;^:^Z;r:Eb<=r:AZ;fZ;FZ;NZ;j:Dk;Ip;xj
:V[;NZBH@]B:dj:Dk;;JSF<<J:dJ;TsgDJEq?La_bFtZ<Z;>d:Xb[og]gffk=v:;J:::::
j:<Jyyy;d:yayY;AZ:>:;P:<:TNC>:UTRcETcTX[US;SK]UW=EWMuUWm>j\\@A:::cYH_W
V;Z::::::::jysyA:CB:F:;jysy?:;:Wt:F]?JTD:=j>r:oR:N:eZ:>:EJ:F[<J:<J<J;@
j:f:FZ<vyyyy:f:F;B:C:WS:kj<jRDj:B:E:Qb:<Z:f:^[<>n>r:WJ:n^@v;_jysy;B:K:
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xY:>Z::::::j<B:sd:B:N:Y<>D_=a=[;;B:::::::JF>:yay=J:B::::::nYyA<:::::::
:::::jysy:>:<:::::::::::::::::::vYxI:;Z:::::::::eB:<jwEJ:uI<:[V:=:;b:<
Kd`ppPps<LaPpu<Lcxpp@Pqf<CJ:f_;j><J:<JSdjHAJmG:;B:;:::::::::::AE:Ko>JS
fu=^T=:A=eJ:VY;syB:>l;>:<Z<:::fICJ:f?=J<JrA:MJ:N@uY:mU:=J:<jR<JrF;N@uC
;eU:=B:B:[MBB:qQ:pI<:[V:B:DZ:nb_og]gfmS@C:UK:^:>X=j>>:_c<;X:;l:<Z:^C[F
:MZ:N@yC;;l:Ja<JBG;<JSF`=>B=:UD:[n>B:_KZG]<F:;ja>l:F;<JSvAmb:=:]LBG;<J
SF@mb:=:[LBO[:B:G;OjyyiyI:;J:><<jw?<IZ:>:sg:>Z:FZ<NZ\\VdsWhngfgcUC:Uk:
^:>X;j;<Z:>:::::::::3:" }}{PARA 0 "" 0 "" {TEXT -1 50 "is an eigenvect
or for any values of the constants." }}{PARA 0 "" 0 "" {TEXT -1 78 "As
 each eigenvector is a directrix there is an infinite number of direct
rices." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 162 "Notice that  the matri
x is diagonal. Whenever the matrix is diagonal there will be an infini
te number of eigenvectors and hence an infinite number of directrices.
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{MARK "27 0 0" 13 }{VIEWOPTS 1 1 0 1 1 1803 }