{VERSION 3 0 "IBM INTEL NT" "3.0" }
{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 16 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 }
{CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 
"" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 
1 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }
{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 
"" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 
1 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 
5 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 20 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 "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 "" 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 "" 0 257 1 {CSTYLE "" -1 
-1 "" 0 1 177 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 "" 0 258 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 "" 0 259 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 }}
{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Matrix Algebra worked exam
ple 4." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "For the systems" }}
{PARA 256 "" 0 "" {OLE 1 4608 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:j::>:B>N:
F:nyyyyy]::yyyyyy:::::::::::::::::::::::::::::::::::::::::::::::::::::
::::::::::::::::::fyyyyya:nYf::G:I:wAyA:::::::::::::::::::::::::::::::
::::::::::::::::::::::::::::::::::::::::::::::::NDYmq^H;C:ELq^H_mvJ:::
:::::gj<vyyuyJ<j?V:;J:>R<:T><::[Xi?wgoIewW:A:;B<B:F:YLpfF>:::::::::J?N
Z;vyyyyyY:vYxY:B:::::::c:;:=:jR>@Wlj^HMMufF;J:::::::N=?:xI:;Z::::::j:>
:;T:B:F:AlqfG[maNFO=;::::::::_J;vyyuy:>:<::::::AZ:^E>:nYN:<j>J<>:wAG:I
Z:>;N;;jCj?>:S:UJ:n;v;;JB<Z:F<N<;jDJDJyKEjEJFjFJG>:q:s:u:w:y:wAyA:::::
::::::::::::::::::::::::::::::::::::::::::::::::::::::::::F:DZ:B::::::
::::^a;J<@Z=fZD^Z;FZ;N\\=Hf<NjcN`=@i@N:?b[uc:KFG_rZke:GvD?B\\_c>AVX?b[
Uc:KfF?RZ;c:=FB:::::::::Z\\OZ;H_lVJG:F:wyyAb<vYxy?V:>:;`:Z@O<;j`@Pt\\P
d`QrPPJPnrPqjLqnxPqF;fbk;::JtaMSAA;B::::::::vYxy;J<<:=:yayQ:>:FcUjQM:c
W;<j:F;HJgE:?JHB:;j<>Z:F[<J:<j<J;@j:j<j:Djyyyy=j<j>Z:f:n_;>=f:F`<F:f:V
[<Z:f:^[<>b>>rRn;n^@v;_jysy;B:K:;xyMxy;F@MB;QZ:n>^;UTRZ_?:;q<B:?B:YLpJ
bNHEms>@[;;B:::::::JF<J;@:<J;>:yayA:;B::::::f:<JqQ:<:?ja:[LsjsJRJ:<:::
::::>=;jysy:>:<::::::wqy[:::::::::::::vYxI:;Z::::::::::::::::::::yay=J
:B::::::::F;B:sZ:Vy<>jxM:<:[V:=:;b:<Kd`ppPps<LaPpu<Lcxpp@PqVuvgwq?taww
qogCN[rJ?^:?b[U;_Jv<j:Dj:=<;;bFAVXGb:RI?b[Uc>AFX?ZLVjsO:Qc:N:;c::J:f_;
B:MB:;B:_c<iT:ck:>Z:Ffs?RM:_;KW:=:kUXOo>JSJ\\Mj:>:k=JSjXYj:jv`QO;_;wx>
=J:>I:_;ex?=:k=JSvl=>r;F:FI:_;uI;=:k=JSJRXj:jv:N@FtMF:FI:_;cx?=:m=s:qA
B:>l;>:<Z<::::jIUJ?`GO:;B:Uk:^:>x;>:MJ:<JSvDGR:=J:<jN`Q:_=VYZ:JBA:;b:D
J:<k`LOJPNvLQs@P:`J?`McmqpYqgmqp?TnG`M:N;jP>:C:[q:F[:>Z:N@QT:SX:=J:<JM
`Q:G;N@NZNF:<JMj>JSVk=nu;F:<JMj>JSjqIk:<Z:n>^=VY;><:[V:>Z<b:<kc`a\\wgf
kIUJ?dWcmqpWU>TvM>:O:f?=J<Jr?:MJ:N`DvDkg:=J:<jX`qcG[:JSjXQj:jO`qc;_;eG
<=J:f_sWTJSJqMk:>^<FjZ::::::::::N@T:TBN:D;;Z]q^E`ZpVa@^^gfhJFdPwdy\\::
:::jX:N@^^PF:;jP:N@iNbEj:<:u;JSJ=Lj:>:U;JSJuQj:<:Q;JSjN\\j:jO:N@FvNF:f
A:_;ib?=:U;sZ:VYZ:JBA:<Z<>:c\\_;::JqbH:f_;>:C:[q:F;;JSvDGy:=B:<JT`QZ=:
N@Vt@F:RX;@b<NsCFZ:Zs;b<vxPF:RX:djEwaCF:RX:d:Oi?=:`AgJ:b<iLHeJ?B:;CB;;
Jj<Jj:::gZ:N@^eHN;>>;;Jj>F::J@<J:N@^]LV:>^\\G_TAF;P:g;N@iNZ^:>^Z>>;Mj:
gZ:N@F\\DN;>>;;Jj>F::JEB:_;]X=O:;KJ:>F;=::><<JSjBAK=B:;KJJj>F:N;;JMT:O
jyyiyI:>:sZ:VY[j=B:;JXE:<j:NZ\\VdsWhngfgsW]U<GENIJ?A:OJ=B:I:J<jPF:C:[Q
:VZ:B:;::::::5:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "find  a)  the \+
equilibrium point" }}{PARA 0 "" 0 "" {TEXT -1 27 "       b)   the eige
nvalues" }}{PARA 0 "" 0 "" {TEXT -1 71 "       c) the eigenvectors\n  \+
     d) the equations of the directrices\n\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 \+
equations" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 13 "X:=4*x+2*y-2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
\"XG,(%\"xG\"\"%%\"yG\"\"#!\"#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "Y:=-x+y+2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"YG,
(%\"xG!\"\"%\"yG\"\"\"\"\"#F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 257 30 
"To find the equilibrium points" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 23 "solve(\{X=0,Y=0\},\{x,y\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#<$/%\"yG!\"\"/%\"xG\"\"\"" }}}{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 258 24 "To find the eigenvalues
 " }{TEXT -1 153 "you first need to enter the matrix A.The first two n
umbers define a 2 by 2 matrix. The four numbers in the square 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 "" {XPPMATH 20 "6#
>%\"AG-%'MATRIXG6#7$7$\"\"%\"\"#7$!\"\"\"\"\"" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 32 "Then use the 'eigenvals' command" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 25 "eigenvalues=eigenvals(A);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#/%,eigenvaluesG6$\"\"#\"\"$" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 38 "there are two real eigenvalues 2 and 3" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT 259 24 "To find the eigenvectors" }{TEXT -1 31 "
 use the 'eigenvectors' command" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 28 "eigensystem=eigenvectors(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#/%,eigensystemG6$7%\"\"#\"\"\"<#-%'VECTORG6#7$!\"\"F(7%\"\"$F(<#-F+6
#7$!\"#F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 173 "The first number 2 \+
is an eigenvalue and the following 1 tells you that it only occurs onc
e, 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 sec
ond 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 th
e value 3." }}{PARA 0 "" 0 "" {TEXT -1 26 "Thus the eigenvectors are \+
" }}{PARA 257 "" 0 "" {OLE 1 4097 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:j::>:
B>N:F:nyyyyy]::yyyyyy:::::::::::::::::::::::::::::::::::::::::::::::::
::::::::::::::::::::::fyyyyya:nYf::G:jy;::::::::::::::::::::::::::::::
:::::::::::::::::::::::::::::::::::::::::::::::::JcvGYMt>^:fBWMtNHm=;:
::::::n<Ejyyiy=C:Qj;>:;JZC:bKB::>h>_nqPyfpq;V:>Z?B:<:=ja^GE=;:::::::::
N;?R:yyyyyyA:yayA:<::::::JDJ:j::F@[KaFFcmnnHEM:>:::::::oJ;Zy=J:B::::::
F:;JBC:j:VBYmp>HYLkNG>::::::::N<?jyyiy=J:B::::::V:<JhJ:JyK;B:M:C:wAG:I
J:<J>J?>:wAQ:S:UJ:n;v;;JB<Z:F<N<;jCJyky;::::::::::::::::::::::::::::::
:::::::::::::::::::::::::::::::::::::::::=Z<B:<::::::::::S>CR:Hj<dj:Hj
;R:ER:=R:?R:FZ<njgNZBP@=b:KVF>Z<nlUOZPhE?RZ;c:=VR?RZWe:=FS?RZ;s:A:E:j:
DJ=kRFZ<>K<J::>:;:::::FZ:nyyYZDjysyQj;B:;JZC:bK_J:fc[_hb_ds?h_?^?GhoGf
nGgioGM:ETV:::^xsNpk[::::::::jysyA:CB:F:;jysy?:;:Yb<F]?JVH:=j>r:sV:NZ:
f<<J:f:F[<Z:^:NZ;F:E:=b:yyyyI:E:M:<J<JQ@JFf:F`<F:<j<j?DZ:B:E:Sb:;G;;S<
W:Gc;YJCvYxY:<J>JZyyKyYj]j[Lj?JMJ@fc[;>:[q<B:M:?B:YLpJbNHEms>@[;;B::::
:::JFNZ;J:N:;B:yayA:;B::::::f:<JjH:<:?ja:[LsjsJRJ:<:::::::>=;jysy:>:<:
:::::wqy[:::::::::::::vYxI:;Z::::::::::::::::::::yay=J:B::::::::JB<Z:V
y<>jx]:JBA:DZ<B>aTXUeRYEUHL<jP@:MB:;B:_c<uV:sp:B:UC:kPBB:Jq>:;:cF=KM:>
::::::::VY;RyB:>l;Z:b:<J]PPqHplHaE^:f_;F:C:[Y:F;;JS>r;>B=B:<J`>LMZ:N@y
C;;l:Ja>LMZ:N@]s:;l:j`<JBG;<JS>Rye:=J:vC[F:MZ:N@ykyDj:jb>LMZ:N@]kyDj:J
b>L[Z:VY;><:[V:b:DZJVDvGhJ<jP>:C:[q:F;;B:_c<unXUj:>Z:f_::<jw?:B:>l;Z:b
:<::jP@j:^:>X;j>>:_KZG^@F:<J`j>B:_kYF^@F:nCF;<JSF`=F^@F:fCF;<JS>RsC<=:
Y<MZ:N@yKX\\j:jbj>B:_kR^aBF:>Df\\:B:qQ:[:JBA:<Z<B>cTTPps<LaPpu<Lcxpp@P
qFPC:UK:^:>X=j>>Z:N@uNTAj:>Z:B:;p>ZDfu=Nc;F:F`:>RJSfMev;=:];JSfUmr;=:]
;YB:;JS>r<v^=f:>^[og^?>;E:KOT?r:>FOZ:n>N;yyyxy:J:><<jw?<IZ:>:sg:>Z:FZ<
NZ\\VdsWhngfgcUC:Uk:^:>x;>:AB:<J::::::::::::::::::::::::::::::::::::::
:::::::::::4:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 260 33 "The equ
ations of the directrices " }{TEXT -1 4 "are " }}{PARA 258 "" 0 "" 
{OLE 1 5121 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:j::>:B>N:F:nyyyyy]::yyyyyy:
::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
fyyyyya:nYf::G:I:K:wAyA:::::::::::::::::::::::::::::::::::::::::::::::
:::::::::::::::::::::::::::::::JcvGYMt>^:fBWMtNHm=;:::::::n<Ejyyiy=C:Q
j;>:;JZC:bKB::>h>_nqPyfpq;V:>bCZ:j:vCSmlJ::::::::::OJ;@jyyyyyy;jysy;Z:
::::::^<>:F::]KRnC=MtFGgml>:;::::::JGN:ry:>:<::::::=J:^^;Z:j:VBYmp>HYL
kNG>::::::::N<?jyyiy=J:B::::::V:<JhJ:JyK;B:M:CJ:nYn:v:;B:K:OJ:>=V;;J@j
@>:W:YJ:>\\:B:]:_J:V<^<f<n<;jEJG>:m:wAq:s:u:w:y:;C:<jJ<:?;A;CC:nYvY:::
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::F:D:<::::::::
::;S:^Z;r:Eb<CR:=R:Ar:FZ?FZ;FZ<F:@JCHRvFZLRXb:gvP?b_uL;TrXDk;ip;H:P::?
b[Uc:KfF@J;\\rRB::rDAr:R;:@J;j:DJ=kR>Z::B:=r:=Jyyy;d:yayY;AZ:>:;`:Z@O<
;j`@Pt\\Pd`QrPPJPnrPqjLqnxPqF;fbk;::JtaMSAA<::::::::yayY:^Z:j:<:yayQ:>
:NaGjdd:cr;F:MZ=Nk=J;<jDJ:f:;j>D:<J<J;@j:f:FZ<vyyyy:f:F;B:C:WS:kj<jRDj
:j<j?DZ:B:E:Sb:;U<lJAJMTjAN<yayAZ:>;>ryYnyArEVZCV;n>^;UTR:;JJDj<B:M:?B
:YLpJbNHEms>@[;;B:::::::JFNZ;J:N:;B:yayA:;B::::::f:<JXX:;B:N:Y<>D_=a=[
;;B:::::::JF>:yay=J:B::::::nYyA<::::::::::::jysy:>:<::::::::::::::::::
:vYxI:;Z:::::::::eB:<jwEJ:uI<:[V:=:;b:<Kd`ppPps<LaPpu<Lcxpp@Pq>@C:US:F
[:>Z:N`Dfm;NX=B:AB:@j<@:B:=b>\\AFINZP<DZw:j<:b:G>F:K^:>Z:>IKG:M:_khIJv
GZ:>:mMN;_KZEjkIj:>:QMN;_khTJsG:>I:_kH]JvG:FI:_K:YjkIj:joJBB:qQ:pY:>Z:
JBAJ:B:DJ:<J]PPqHplHaR^:f?=J<JrA:MJ:N@;X:;l:<Z:^C[F:MZ:N@yC;;l:Ja>LMZ:
N@]s:;l:j`<JBG;<JS>RiW:=J:vC[F:MZ:N@ykUAj:jb>LMZ:N@]kUAj:Jb>L[Z:VY;><:
[V:b:DJ:<kc`a\\wgfC@C:UK:^:>X=j>>Z:N`D>r<Fo<F:;B:uKfG;N@uV:uT;=J:f_:>U
[Z:VY;:<:[V:B:DZ:::f_;F:C:[Q:F;;JS>r;Fr>F:;B:S<MZ:N@ykZMj:Jaj>B:_kRHjZ
Mj:j`j>B:_KZ?ZAF:vCF;<JSvA;r;=:]LBG;<JSF@;r;=:[<[B:<jw?JB:>l;Z<b:<kcjq
:f?;J<JrG:MJ:<JSdJZEjgYj:B:QKfG;N@upFaj:jP<:[Z:Vy<>:B:>l;Z:b:<::jP@j:^
:>X;j>>:_KZAjx]j:B:S<MZ:N@ykx]j:Jaj>B:_kRHjx]j:j`j>B:_KZ?]EFZ:jaj>B:_k
Y>]EF:FDF;<JSF@kr<=:[<MZ:N@;f;;l:<:S<MZ:N@yI<;l:Jaj>B:_krYJZF:fCF;<JS>
JiW:=J:vCF;<JSvYiW:=:]<MZ:N@]qUAj:JbJBB:qQ:[:JBA:DZ<B>jq:f?;J<JrG:MJ:<
JSdJ:YJ:DJE;V?j<:<j:D:@Z@<<J:DJE;;J::::::::::::::Fo<F:fa:j>JSfe@fe?F:;
jP<:[Z:VY;J:B:>l;Z:b:;B:::US:=J<Jr?:MJ:N@;f;=H;=Z:^CF;<JSvY=H;=:W<MZ:N
@]qZMj:j`j>B:_K:_u@F:;jaj>B:_ky_u@F:FD[n>B:_kr_u@F:>D><<jw?JB:>l;Z<b:<
kcjq:f?;J<JrG:MJ:<JSdJ:YjUYj:B:Q;M:_kH]jt`j:jP<:[Z:VY;:<:[V:B:DZ:::f_;
F:C:[Q:F;;JS>j@FuBF:<J`j>B:_kyGuBF:nCF;<JSFXmH<=:U<MZ:N@;Ntej:jaj>B:_k
y_xDF:FDF;<JSFXci<=:[<eB:<jw?JB:>l;Z:b:<Kd``_;ND:C:UK:^:>X=j>>Z:N@uV:e
c;=Z:B:;p>ZDfUQX;=J:F`:>RJSfMeY<=:];JSfU]e<=:];JSfe@f`@F:F@:_kH]jVQj:j
R:N@ulbaj:jR:N@un`dj:<:];OZ:n>N;yyyxy:J:><<jw?<IJ:<J:^q<J:<j:NZ\\VdsWh
ngfgcUC:Uk:^:>X;j;<Z:>:::::::::::::::::::::::::::::::::6:" }}}{EXCHG 
{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 24 "b)   Consider the system" }}{PARA 256 "" 0 "" 
{OLE 1 3595 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:j::>:B>N:F:nyyyyy]::yyyyyy:
::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
fyyyyya:nYf::wyyyqy;::::::::::::::::::::::::::::::::::::::::::::::::::
::::::::::::::::::::::::::::::NDYmq^H;C:ELq^H_mvJ::::::::gj<vyyuyJ<j?V
:;J:>R<:T><::[rlEwgoIewW:A:;B;B:F:YLpfF>:::::::::J?NZ;vyyyyyY:vYxY:B::
:::::c:;:=:jR>@Wlj^HMMufF;J:::::::N=?:xI:;Z::::::j:>:C=;:=j[vGUMrvC?Mo
J::::::::JCN:yyyxI:;Z::::::j;B:s<;:wA?Z::C:J=j=B:K:M:OJ:V;;J@j@>:W:YJ:
nYvY::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
::::::::F:DZ:B::::::::::>F<J<@Z=fZD^Z;FZ;N`=@i@N:?b[uc:KfF_rZkE:=bFAVX
Gb::RX;Z:bIP\\lb=X:@Jb<RMt><::::::::::=B:wyyAb<vYxy?V:>:;`:Z@O<;j`@Pt
\\Pd`QrPPJPnrPqjLqnxPqF;fbk;::JtaMSAA;B::::::::vYxy;J<<:=J:vYxY;J:JuM:
YG;nf<B:=j>r:aS:<J;f<>:EJ:F[<J:<J<B:?R:=j<j:Djyyyy=j<j>Z:^:n_;>=E:]c:=
Z:f:V[<Z:f:^[<>b>>j>n;;JMTjAN<yayAZ:>;>ryysyAb>v[>V;n>^;UTR:;Jj=j<B:M:
?B:YLpJbNHEms>@[;;B:::::::JFNZ;J:N:;B:yayA:;B::::::f:<JAE:<:?ja:[LsjsJ
RJ:<:::::::>=;jysy:>:<::::::wqy[:::::::::::::vYxI:;Z::::::::::::::::::
::yay=J:B::::::::jBjwEJ:uI;B:>l;B:DZ<>ZJfcN?^??inOh[s=C:US:F[:>Z:N`Dfb
;>H=B:<JM<J<G;N@Ev:gm:JMjDB:qQ:[:JBAj:J:D:cTTUUSaEBWTSiEB_tUUURWmx^:f?
=J<JrA:MJ:N@]T:Sl:B:kMN=j>JSJnHj:Jv:N@]v:kl:<:mMN;_;;v:=B:FI>\\:B:qAB:
>l;>:DZ<B>aTXDpql`S^:f?;J<JrG:MJ:N`DFDsT:=J:<jX>UM:_kB?g;F:fAf<VYZ:JBA
:<Z<JdRS::SK<<:Uk:^:>X;j>>:_kbNk<F:;B:<JZ=j>ZDFLgf:=:<:OZ:n^@J?vyyuy=:
;JBB:qQBv:>:sg:B:=b:?bBaTXaEWEUUtP<jP>:C:[q:VZ:B:;::::::::::6:" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "find a) the equilibrium point" }}
{PARA 0 "" 0 "" {TEXT -1 93 "      b)  the eigenvalues\n      c) the e
igenvectors\n     d) the 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 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 261 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 equilibrium point is obv
iously (0,0)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT 262 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 "Thu
s the eigenvalue is 2 twice. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 256 24 "To find the eigenvectors" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "eigensystem:=eigenvectors(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 follows. Let" }}
{PARA 257 "" 0 "" {TEXT -1 1 " " }{OLE 1 3595 1 "[xm]Br=WfoRrB:::wk;ny
yI;G:;:j::>:B>N:F:nyyyyy]::yyyyyy:::::::::::::::::::::::::::::::::::::
::::::::::::::::::::::::::::::::::fyyyyya:nYf::wyyyqy;::::::::::::::::
::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::NDYmq^
H;C:ELq^H_mvJ::::::::gj<vyyuyJ<j?V:;J:>R<:T><::[rlEwgoIewW: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>J?>:wAQ:S:UJ:n;v;;Jyky;::::::::::::::::::::::::::::::
:::::::::::::::::::::::::::::::::::::::::::j:b:<Z::::::::::J:C:CR:Hj<d
j:Hj;R:ER:=R:?R:FZLVj]WZIF:QR:?B\\]kB<ZDFZLVJ:N@]Z:>ZDNZPXe<nlW[nSCD]H
b:R:[DZATRWuREEUv:IJ:>:::::FZ:nyyYZDjysyQj;B:;JZ?Z:Z@O<;j`@Pt\\Pd`QrPP
JPnrPqjLqnxPqF;fbk;::JtaMSAQ:B::::::::vYxy;J<<:=J:vYxY;J:JaH:mR;^`<j:F
;HJG@:?jDB:;j<>:Mb:>Z:^:NZ;F:E:=b:yyyyI:E:M:<J<JQ@JFf:F`<F:<j<j?DZ:B:E
:Sb:;G;HJA>:Gc;YJCvYxY:<J>JZyyKyYZEj[Lj?JMJ@fc[;>:[q<B:M:?B:YLpJbNHEms
>@[;;B:::::::JFNZ;J:N:;B:yayA:;B::::::f:<JhD:<:?ja:[LsjsJRJ:<:::::::>=
;jysy:>:<::::::wqy[:::::::::::::vYxI:;Z::::::::::::::::::::yay=J:B::::
::::jD<Z:Vy<>jx]:JBAj:J:DZJ^dcgg_WhZnc_whZNdigg[oGeJ<>:US:F[:>Z:N`Dfm;
nV=J:<J:::::::::::jk<JNG;N@ux:cp:jkjD>:qQJxI<:[V:;Z:b::::uM<>:Uk:^:>x;
j>>:_kxAjv@j:>Z:F`:>HM:_kXLjt@j:<:<Jr><<jw?Zw]:JBA:<Z<B:GTSWuREuV`K<jP
>:C:[q:F;;JSdJZAJZFZ:B:SLB=j>B:_kYLJZF:nc:>LMZ:N@]s:;l:j`<JBG;<JS>Rmb:
=J:vC[F:MZ:N@ykFDj:jb>LMZ:N@]kFDj:Jb>LOB:<JMJ?vyyuy=J:>:[Z:VY[j=B:;JXE
:;B:=b:?bBaTXaEWEUUtP<jPF:C:[Q:VZ:B:;::::::::4:" }}{PARA 0 "" 0 "" 
{TEXT -1 19 "be any vector. Then" }}{PARA 258 "" 0 "" {OLE 1 5131 1 "[
xm]Br=WfoRrB:::wk;nyyI;G:;:j::>:B>N:F:nyyyyy]::yyyyyy:::::::::::::::::
::::::::::::::::::::::::::::::::::::::::::::::::::::::fyyyyya:nYf::G:I
:K:wAyA:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
:::::::::::::::JcvGYMt>^:fBWMtNHm=;:::::::n<Ejyyiy=C:Qj;>:;JZC:bKB::>p
KwnqPyfpq;V:>bCZ:j:vCSmlJ::::::::::OJ;@jyyyyyy;jysy;Z:::::::^<>:F::]KR
nC=MtFGgml>:;::::::JGN:ry:>:<::::::=J:^b;Z:j:VBYmp>HYLkNG>::::::::N<?j
yyiy=J:B::::::V:<JhJ:JyK;B:M:CJ:nYn:v:<J>j@>:OJ:V;;J@JyKAjA>:[B:<jBJC>
:a:c:e:gJ:v<>=F=N=;jGJHjHJIjIJJ<Z:F^:JKjKJLJyky;::::::::::::::::::::::
:::::::::::::::::::::::::::::::::::::j:b:B::::::::::>b;J<@Z=fZDFZ=V:@j
<@j:@J;D:=b:KfF?b[Uc:CFU@J;>@=:JS:>:?b_qd:gVQHj;j<Z;j:Dk;Ip;xj:V[;NZBH
@]B:dj:Dk;;JS<jBB:;b<?b_q<DJ=qm;Hj:nyyYZDjysyQj;B:;JZC:bK_J:<j`@Pt\\Pd
`QrPPJPnrPqjLqnxPqF;fbk;::JtaMSAA<::::::::yayY:^Z:>:=B:vYxY;J:jJa;mR;n
n@j:F;HjcH:?B:eZ:>:EJ:F[<Z:^:NZ;F:E:=b:yyyyI:E:M:<J<JQ@JFf:F`<F:f:V[<B
:<j<J@DJJMJZXKAJMTjAN<yayAZ:>;;JZyyKyiQVb>V;<JMJ@fc[;>:[q<B:M:?B:YLpJb
NHEms>@[;;B:::::::JFNZ;J:N:;B:yayA:;B::::::f:<JaU:;B:N:Y<>D_=a=[;;B:::
::::JF>:yay=J:B::::::nYyA<::::::::::::jysy:>:<:::::::::::::::::::vYxI:
;Z:::::::::eB:<jwEJ:uI<:[V:<Z:b:<Kd`ppPps<LaPpu<Lcxpp@Pq^GCB:f_;j><J:<
JSdjHAJdG:<j:Pj:b:?Z;NZB@@=::Z:>Z<nLj;j:P:F:;R:_::@=;J?@JC:ZlR>>Z:N[<:
N:u[:j<J;j:DZjR:J:<J;:<Z;vH?c:=B>[Ke=KLD:;B:gDB[K:<JeGB;B;B:::::::::::
:::::_C:;p>ZDjoTj:>Z:>@;H:JSfu=fy;F:>@:_;?T;=:<:ZDfMcV==:[;JSJRyj:>:[;
JSfUcV==:[;JSJRyj:JR:N@u^OFZ:JR:N@u`OF:>@f\\:jw?^y]:JBA:<Z<>:c<::uM<jP
J<JrA:MJ:N@uY:cc<=Z:F`:>HM:_kXLJRdj:<:<Jr:djx?dLF:;jR:N@uK`Dk:Z:JBB:qQ
:[:JBA:DZ<B>aTXDpql`S^:f?;J<JrG:MJ:N`DfMkX:=J:<jP<JfG;N@uP]Tj:jPJBB:qQ
:pI<:[V:B:DZ:nb_og]gfmS@C:US:=J<Jr?:MJ:N@;X:;l:<Z:^C[F:MZ:N@yC;;l:Ja>L
MZ:N@]s:;l:j`<JBG;<JS>RaD<=:YLB=j>B:_kYVdBF:FD[n>B:_kRVdBF:>D[n>B:_KZW
yBF:^CF;<JSvAqI<=:W<MZ:N@]kw]j:j`j>B:_KZ_tDF:vCF;<JSvAch<=:]<MZ:N@]Kde
j:JbJBB:qQ:[:JBA:DZ<B>jq:f?;J<JrG:MJ:<JSdJZEJ]ij:>Z:fAk@<jw?:B:>l;Z:b:
<::jPF:C:[Q:F;;JSR:;R==Z:^CF;<JSvA;R==:W<MZ:N@]s:;R==:U<MZ:N@;pq<;;jaj
>B:_kYvgJF:FDF;<JSF@YE>=:[<MZ:N@;pE@k:>:S<MZ:N@ykE@k:Jaj>B:_kRv\\KF:fC
F;<JS>R[i>=J:<jaj>B:_kY>xL:]<MZ:N@]KrEk:JbJBB:qQ:[:JBA:DZ<B>a<Y=jP>:C:
[q:F;;B:_c<;h:?y>=J:<jX:<jw?:B:>l;Z:b:<::jPF:C:[Q:F;;JSR:sR?=J:<J`j>B:
_kY^]OF:nCF;<JSF`=^]OF:fCF;<JS>RYe?=:Y<MZ:N@ykqTk:jb>LMZ:N@]kqTk:JbJBB
:qQ:[:JBAj:>Z<b:<;vG:UK:^:>X=j>>Z:N`DfM[i:=B:;B:SMfG;N@uPiXj:JpjDB:qA>
Z:JBA:;J:DJ:^DP@Jc:SM<jPF:C:[Q:F;;JSfM=W<=Z:Vf:>OM:_khwkCF:VF:_kHgqKF:
;jk:N@upOAk:jkJ?B:G;OjyyiyI:>:[Z:VY[j=B:;JXE:;B:=b:?bBaTXaEWEUUtP<>:UK
:^:>X=j;<J:<J:::::::::::::::::::::::::::::::::::::::::::4:" }}{PARA 0 
"" 0 "" {TEXT -1 17 "Hence the vector " }}{PARA 259 "" 0 "" {OLE 1 
3599 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:j::>:B>N:F:nyyyyy]::yyyyyy::::::::
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::fyyyyya
:nYf::wyyyqy;:::::::::::::::::::::::::::::::::::::::::::::::::::::::::
:::::::::::::::::::::::NDYmq^H;C:ELq^H_mvJ::::::::gj<vyyuyJ<j?V:;J:>R<
:T><::[WNIwgoIewW:A:;B;B:F:YLpfF>:::::::::J?NZ;vyyyyyY:vYxY:B:::::::c:
;:=:jR>@Wlj^HMMufF;J:::::::N=?:xI:;Z::::::j:>:C>;:=j[vGUMrvC?MoJ::::::
::JCN:yyyxI:;Z::::::j;B:s<;:wA?Z:F;^:nYn:v:<J>J?>:wAQ:S:UJ:n;v;;Jyky;:
::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
::j:b:<Z::::::::::J:C:CR:Hj<dj:Hj;R:ER:=R:?R:FZLVj]WZIF:QR:?B\\]kB<ZDF
ZLVJ:N@]Z:>ZDNZPXe<nlW[nSCD]Hb:R:[DZATRWuREEUv:IJ:>:::::FZ:nyyYZDjysyQ
j;B:;JZ?Z:Z@O<;j`@Pt\\Pd`QrPPJPnrPqjLqnxPqF;fbk;::JtaMSAQ:B::::::::vYx
y;J<<:=J:vYxY;J:JaH:mR;^`<j:F;HJG@:?jDB:;j<>:Mb:>Z:^:NZ;F:E:=b:yyyyI:E
:M:<J<JQ@JFf:F`<F:<j<j?DZ:B:E:Sb:;G;HJA>:Gc;YJCvYxY:<J>JZyyKyYZEj[Lj?J
MJ@fc[;>:[q<B:M:?B:YLpJbNHEms>@[;;B:::::::JFNZ;J:N:;B:yayA:;B::::::f:<
JhD:<:?ja:[LsjsJRJ:<:::::::>=;jysy:>:<::::::wqy[:::::::::::::vYxI:;Z::
::::::::::::::::::yay=J:B::::::::jD<Z:Vy<>jx]:JBAj:J:DZJ^dcgg_WhZnc_wh
ZNdigg[oGeJ<>:US:F[:>Z:N`Dfm;nV=J:<J:::::::::::jk<JNG;N@ux:cp:jkjD>:qQ
JxI<:[V:;Z:b::::uM<>:Uk:^:>x;j>>:_kxAjv@j:>Z:F`:>HM:_kXLjt@j:<:<Jr><<j
w?Zw]:JBA:<Z<B:GTSWuREuV`K<jP>:C:[q:F;;JSdJZAJZFZ:B:SLB=j>B:_kYLJZF:nc
:>LMZ:N@]s:;l:j`<JBG;<JS>Rmb:=J:vC[F:MZ:N@ykFDj:jb>LMZ:N@]kFDj:Jb>LOB:
<JMJ?vyyuy=J:>:[Z:VY[j=B:;JXE:;B:=b:?bBaTXaEWEUUtP<jPF:C:[Q:VZ:B:;::::
:::::2:" }}{PARA 0 "" 0 "" {TEXT -1 50 "is an eigenvector for any valu
es of the constants." }}{PARA 0 "" 0 "" {TEXT -1 78 "As each eigenvect
or is a directrix there is an infinite number of directrices." }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 162 "Notice that  the matrix is diagon
al. Whenever the matrix is diagonal there will be an infinite number o
f eigenvectors and hence an infinite number of directrices." }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}}{MARK "44 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }