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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]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "\nIt i
s easy to get the eigenvalues, but watch out, they could be complex nu
mbers." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "eigenvals(A);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 474 "\nIf you request the eigenvectors you ac
tually get more: a LIST (or a SEQUENCE of LISTS) in which the first en
try is the eigenvalue, the second is the multiplicity (typically in re
al life problems this will be one), and the third is a SET of independ
ent eigenvectors that belong to the eigenvalue in question.  In our fi
rst example there is a repeated eigenvalue (-3/2 appeared twice), so t
he multiplicity is 2, but there is only one independent eigenvector,  \+
namely [1,  1]." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "V:= eigenvects(A
);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 302 "\nTo extract the eigenvect
or use the op command.  We want the first (and only) operand of the th
ird operand of V. (Think of this as stripping off the large square bra
ckets and selecting the third item that lies therein, then stripping o
ff the curley braces and taking the one thing that is inside them.)" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "v1:= op( op(3, V) );" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 143 "Check fo
r yourself that this is correct!  Is it true that A v1 =  (-3/2) v1 ? \+
 Let's try using our transform procedure to do a graphical check." }}}
{SECT 1 {PARA 261 "" 0 "" {TEXT 257 24 "Transformation procedure" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 143 "transform:= proc( A :: matr
ix , 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 );" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 482 "\nHere's another example, this time with two distinct complex \+
eigenvalues.  Oftentimes we don't care about specific complex eigenval
ues; the mere absence of real eigenvalues is all we need to know.  Oth
er times we only care about features of the complex eigenvalues, such \+
as whether they are pure imaginary, or if they have positive or negati
ve real parts.  For this sort of thing, see E-K section 4.7, figure 4.
4 on p. 137, section 4.9, and figures 5.11 and 5.12 on pp. 185 and 187
." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "B:= matrix(2, 2, [4, 1, -2, 3]
);   eigenvals(B);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 57 "The next one has two distinct pure imaginary ei
genvalues." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "C:= matrix(2, 2, [-0.
4, -4, 2, 0.4]);   eigenvals(C);\n " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "The next has two distict real e
igenvalues." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "DD:=  matrix(2, 2, [
2, 3, 4, 1]);   eigenvals(DD);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 19 "V:= eigenvects(DD);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 366 "
\nLet's see if we can extract those two independent eigenvectors from \+
this mess.  First we turn the SEQUENCE called V into a LIST by putting
 it in [ ].  Then we dig in.  The first eigenvector is the first (and \+
only) item of the third item of the first item of [V]; the second is t
he 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  " }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 693 "How about that?  \+
Fortunately, unless we are planning to write down explicit solutions o
f our differential equations, we seldom need all this.  And since the \+
solutions corresponding to the various eigenvectors are solutions of t
he linearized system, which we are only using as a guide to the behavi
our of the original non-linear systems in the vicinity of equilibrium \+
points, there isn't a lot of point in getting formulas.  All we are re
ally interested in is the stability properties and local behaviour of \+
the linearized system (which is supposed to reflect that of the origin
al system), and, as mentioned above, this is often possible to read ju
st from the kinds of eigenvalues that we get. " }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart;" }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 
1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }