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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 1 " " }{TEXT 256 9 "eigen.mws" }{TEXT -1 58 "   Linear
 algebra: eigenvalue and eigenvector calculations" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "restart:   with(lin
alg):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 488 "Don't worry about the warnings--the standard meanings of
 norm and trace in Maple have to do with number theory, so this is jus
t reminding us that we are going to be using the linear algebra meanin
gs for now.  (If you are doing number theory and linear algebra simult
aneously, then you are probably smart enough to get around this little
 conflict.) \n\nA matrix is entered by giving the number of rows, numb
er of columns, and a list of the entries from left to right, top row t
o bottom row. " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "A:= matrix( 2 , 2
, [ -4, 5/2, -5/2, 1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AGK%'ma
trixG6#7$7$!\"%#\"\"&\"\"#7$#!\"&F-\"\"\"Q(pprint06\"" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 81 "\nIt is easy to get the eigenvalues, but \+
watch out, they could be complex numbers." }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "eigenvals(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$#!\"
$\"\"#F#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 474 "\nIf you request the
 eigenvectors you actually get more: a LIST (or a SEQUENCE of LISTS) i
n which the first entry is the eigenvalue, the second is the multiplic
ity (typically in real life problems this will be one), and the third \+
is a SET of independent eigenvectors that belong to the eigenvalue in \+
question.  In our first example there is a repeated eigenvalue (-3/2 a
ppeared twice), so the multiplicity is 2, but there is only one indepe
ndent eigenvector,  namely [1,  1]." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
18 "V:= eigenvects(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"VG7%#!\"
$\"\"#F(<#K%'vectorG6#7$\"\"\"F.Q(pprint06\"" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 302 "\nTo extract the eigenvector use the op command.  We w
ant the first (and only) operand of the third operand of V. (Think of \+
this as stripping off the large square brackets and selecting the thir
d item that lies therein, then stripping off the curley braces and tak
ing the one thing that is inside them.)" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 20 "v1:= op( op(3, V) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#v1G
K%'vectorG6#7$\"\"\"F)Q(pprint06\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 143 "Check for yourself that this i
s correct!  Is it true that A v1 =  (-3/2) v1 ?  Let's try using our t
ransform procedure to do a graphical check." }}}{SECT 0 {PARA 261 "" 
0 "" {TEXT 257 24 "Transformation procedure" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 143 "transform:= proc( A :: matrix , V :: list )\nlocal
 i;\nplot(\{ V, [ seq( convert( multiply( A, op(i, V) ), list), i = 1 \+
.. nops( V ) ) ] \} );\nend:" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 25 "seg:= [ [0, 0], [1, 1] ]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 21 "transform( A , seg );" }}{PARA 13 "" 1 "" {GLPLOT2D 624 624 
624 {PLOTDATA 2 "6&-%'CURVESG6$7$7$$\"\"!F)F(7$$\"\"\"F)F+-%'COLOURG6&
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0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 482 "\nHere's another example, this time with two distinct co
mplex eigenvalues.  Oftentimes we don't care about specific complex ei
genvalues; the mere absence of real eigenvalues is all we need to know
.  Other times we only care about features of the complex eigenvalues,
 such as whether they are pure imaginary, or if they have positive or \+
negative real parts.  For this sort of thing, see E-K section 4.7, fig
ure 4.4 on p. 137, section 4.9, and figures 5.11 and 5.12 on pp. 185 a
nd 187." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "B:= matrix(2, 2, [4, 1, \+
-2, 3]);   eigenvals(B);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BGK%
'matrixG6#7$7$\"\"%\"\"\"7$!\"#\"\"$Q(pprint06\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$,&#\"\"(\"\"#\"\"\"*&^##F'F&F'F%F*F',&F$F'*&^##!\"\"F&F
'F%F*F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 57 "The next one has two distinct pure imaginary eigenvalues.
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "C:= matrix(2, 2, [-0.4, -4, 2, \+
0.4]);   eigenvals(C);\n " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CGK%'
matrixG6#7$7$$!\"%!\"\"F+7$\"\"#$\"\"%F,Q(pprint06\"" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6$^$$\"\"!F%$\"+++++G!\"*^$F$$!+++++GF(" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "The next \+
has two distict real eigenvalues." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
50 "DD:=  matrix(2, 2, [2, 3, 4, 1]);   eigenvals(DD);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%#DDGK%'matrixG6#7$7$\"\"#\"\"$7$\"\"%\"\"\"Q(ppr
int06\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"&!\"#" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 19 "V:= eigenvects(DD);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"VG6$7%\"\"&\"\"\"<#K%'vectorG6#7$F(F(Q(pprint06\"7%
!\"#F(<#KF+6#7$F(#!\"%\"\"$Q(pprint1F/" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 366 "\nLet's see if we can extract those two independent eige
nvectors from this mess.  First we turn the SEQUENCE called V into a L
IST by putting it in [ ].  Then we dig in.  The first eigenvector is t
he first (and only) item of the third item of the first item of [V]; t
he second is the first (and only) item of the third item of the second
 item of [V] (got all that?).  " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "
v1:=op( op(3, op(1, [V]) ));   v2:=op( op(3, op(2, [V]) ));\n  " }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#v1GK%'vectorG6#7$\"\"\"F)Q(pprint06
\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#v2GK%'vectorG6#7$\"\"\"#!\"%
\"\"$Q(pprint06\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 693 "How about that?  Fortunately, unless we are plann
ing to write down explicit solutions of our differential equations, we
 seldom need all this.  And since the solutions corresponding to the v
arious eigenvectors are solutions of the linearized system, which we a
re only using as a guide to the behaviour of the original non-linear s
ystems in the vicinity of equilibrium points, there isn't a lot of poi
nt in getting formulas.  All we are really interested in is the stabil
ity properties and local behaviour of the linearized system (which is \+
supposed to reflect that of the original system), and, as mentioned ab
ove, this is often possible to read just from the kinds of eigenvalues
 that we get. " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}
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