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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 258 28 "Eigenvalues and Eigenvect
ors" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 63 "Part 1.  Maple commands relate
d to eigenvalues and eigenvectors" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "Enter the matrix " }{XPPEDIT 18 0 
"A" "6#%\"AG" }{TEXT -1 47 " .  Calculate the characteristic polynomia
l of " }{XPPEDIT 18 0 "A" "6#%\"AG" }{TEXT -1 41 " by hand before you \+
go to the next step. " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linal
g):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "A:=matrix(3,3,[2,0,0,-8,4,-6
,8,1,9]);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Construct the char
acteristic matrix " }{XPPEDIT 18 0 "lambda*I-A" "6#,&*&%'lambdaG\"\"\"
%\"IGF&F&%\"AG!\"\"" }{TEXT -1 1 "." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
29 "evalm(lambda*diag(1,1,1)-A);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 70 "Compare the result of the preceding step with Maple's charmat c
ommand." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "charmat(A,lambda);\n" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 120 "Take the determinant of the char
acteristic matrix.  Compare with your hand calculation of the characte
ristic polynomial." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "det(%);\n" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "Compare the result of the precedin
g step with Maple's charpoly command." }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 20 "charpoly(A,lambda);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "Th
e characteristic polynomial appears in partially factored form because
 of the zeros in " }{XPPEDIT 18 0 "A" "6#%\"AG" }{TEXT -1 70 ".  Finis
h the factorization by hand, and determine the eigenvalues of " }
{XPPEDIT 18 0 "A" "6#%\"AG" }{TEXT -1 56 " before you go on.  Check by
 asking Maple for the roots." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "roo
ts(%);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 159 "The 1's in each pair
 (second entries) are the algebraic multiplicities of the roots.  The \+
first entries of these pairs should agree with your hand calculation.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "Now l
et Maple compute the eigenvalues directly." }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "eigenvals(A);\n\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 26 "One of the eigenvalues of " }{XPPEDIT 18 0 "A" "6#%\"AG" }
{TEXT -1 103 " is 6.  Let's find corresponding eigenvectors (a basis f
or the eigenspace) by finding the nullspace of " }{XPPEDIT 18 0 "6*I-A
" "6#,&*&\"\"'\"\"\"%\"IGF&F&%\"AG!\"\"" }{TEXT -1 1 "." }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 32 "B:=charmat(A,6); nullspace(B); \n" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 110 "Before you go on, fill in here what the \+
last answer means.  Check your answer with an appropriate calculation.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "Back up a step, and replace t
he word \"nullspace\" with \"kernel\" to see what changes." }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 127 "Now ente
r Maple's eigenvects command, and carefully explain the output.  If an
y part of it looks mysterious, use ? to get help." }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 15 "eigenvects(A);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 56 "Finally, we consider a variant of the eigenvals command:" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "Eigenvals(A);\n" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 175 "The result may not look very exciting.  That's b
ecause Eigenvals is an \"inert\" function.  To see the values it produ
ces, put an evalf() around the command, and enter it again." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "Now enter Eigen
vals with a second parameter:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "ev
alf(Eigenvals(A,P)); print(P);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
21 "Study the columns of " }{XPPEDIT 18 0 "P" "6#%\"PG" }{TEXT -1 40 "
, and explain carefully what the matrix " }{XPPEDIT 18 0 "P" "6#%\"PG
" }{TEXT -1 4 " is." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 65 "Enter the following command, and explain what y
ou think it shows." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "evalm(inverse
(P)&*A&*P);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 21 "Part 2.  The m
atrix P" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 49 "Enter the following line, and explain the output." }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "v2:=col(P,2); evalm(A&*v2); evalm(6
*v2);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 121 "Confirm your observat
ion in the preceding step by carrying out the same operations with the
 first and third columns of P." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 28 "Part 3.  No real eigenva
lues" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 17 "Enter the matrix " }{XPPEDIT 18 0 "B" "6#%\"BG" }{TEXT -1 1 ".
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "B:=matrix(2,2,[1,-1,1,1]);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 23 "Part 4.  Multiplicities
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
16 "Enter the matrix" }{TEXT 262 2 " X" }{TEXT -1 1 "." }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 87 "X:=matrix(4,4,[331,290,-58,580,603,448,-213,10
05,-105,-75,38,-175,-498,-395,143,-847]);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 16 "Part 5.  Summary" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
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