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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "Eigenvalues of linear tran
sformations," }}{PARA 0 "" 0 "" {TEXT -1 39 "and the eigendecompositio
n of a matrix." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "pi := evalf(Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%#piG$\"+aEfTJ!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "Plot a circ
le (as a sequence of N points). " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 130 "N := 60;\np := vector(N+1, t -> [cos(2*pi*(t-1)/N), \+
sin(2*pi*(t-1)/N)]):\nplot( convert(p,list), scaling=CONSTRAINED, styl
e=POINT );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG\"#g" }}{PARA 13 "
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y*F.$!1+++<p6z?F.7$$\"1+++c*=_%**F.$!1+++8YGX5F.7$F($!1+++xkez6!#C-%'C
OLOURG6&%$RGBG$\"#5F^tF*F*-%&STYLEG6#%&POINTG-%(SCALINGG6#%,CONSTRAINE
DG-%+AXESLABELSG6$%!GF_\\l-%%VIEWG6$%(DEFAULTGFc\\l" 2 400 300 300 5 
0 1 0 2 9 0 4 1 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 
0 0 0 0 0 1 1 0 0 0 -27741 16347 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 69 "Procedure to apply a linear transformation A to a seque
nce of points." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "transfor
m := proc(A,p)\n   local i;\n   RETURN(map(point ->\n     convert(mult
iply(A,vector(point)),list),\n   p))\nend:\n" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 103 "A := matrix(2,2, [3,0, 0,1]);\neigenvals(A);\np
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11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7$7$\"\"$\"\"!7$F+\"\"\"" 
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\"1+++9R&e8$F1$!1+++`*=_%**F17$$\"1+++12NPiF1$!1+++4gZ\"y*F17$$\"1+++Q
)40F*F1$!1+++i^c5&*F17$$\"1+++F*4-A\"F.$!1+++!eaa8*F17$$\"1+++++++:F.$
!1+++QSDg')F17$$\"1+++kdNj<F.$!1+++F*p,4)F17$FQ$!1*****fD[9V(F17$$\"1+
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300 300 5 0 1 0 2 9 0 4 1 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 
0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 5 0 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 86 "theta := pi/6;\nRotation := matrix(2,2,[cos(thet
a),-sin(theta),sin(theta),cos(theta)]);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%&thetaG$\"+dx)fB&!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)Rot
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "A1 := evalm( Rotation &* dia
g(3,1) &* transpose(Rotation));\neigenvals(A1);\n" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#A1G-%'matrixG6#7$7$$\"+*******\\#!\"*$\"+TSDg')!#57$
F-$\"+++++:F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+&*********!#5$\"+
++++I!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "plot( transfor
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{INLPLOT "6'-%'CURVESG6$7in7$$\"1+++*******\\#!#:$\"1+++TSDg')!#;7$$\"
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{MPLTEXT 1 0 51 "A2 := randmatrix(2,2);\nmap(evalf, [eigenvals(A2)]);
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(o6Fip7$$\"1+++bN*zc%F0$!1+++Dg&45\"Fip7$$\"1+++*Hr*eXF0$!1+++Z:2@5Fip
7$$\"1+++,+++XF0$!1+++6+++$*F0-%'COLOURG6&%$RGBG$\"#5!\"\"F*F*-%&STYLE
G6#%&POINTG-%(SCALINGG6#%,CONSTRAINEDG-%+AXESLABELSG6$%!GFh_l-%%VIEWG6
$%(DEFAULTGF\\`l" 2 400 300 300 5 0 1 0 2 9 0 4 1 1.000000 45.000000 
45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 3 8236 0 0 0 0 
0 0 }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Remember the " }{TEXT 256 
36 "eigenvalue decomposition of a matrix" }{TEXT -1 1 ":" }}{PARA 0 "
" 0 "" {TEXT -1 56 "A square, nonsingular real matrix A can be decompo
sed as" }}{PARA 0 "" 0 "" {TEXT -1 6 "      " }{XPPEDIT 18 0 "A = X*La
mbda*X^(-1);" "6#/%\"AG*(%\"XG\"\"\"%'LambdaG\"\"\")%\"XG,$\"\"\"!\"\"
F'" }{TEXT -1 13 "     (i.e.,  " }{XPPEDIT 18 0 "A*X = X*Lambda;" "6#/
*&%\"AG\"\"\"%\"XG\"\"\"*&%\"XGF&%'LambdaGF&" }{TEXT -1 2 " )" }}
{PARA 0 "" 0 "" {TEXT -1 44 "where X is a square nonsingular matrix,\n
and " }{XPPEDIT 18 0 "Lambda;" "6#%'LambdaG" }{TEXT -1 50 " is a diago
nal matrix of complex eigenvalues of A." }}{PARA 0 "" 0 "" {TEXT -1 4 
"The " }{TEXT 298 14 "column vectors" }{TEXT -1 21 " of X are called t
he " }{TEXT 297 12 "eigenvectors" }{TEXT -1 6 " of A." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "X := 'X';" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"XGF$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "
eigenvalues := evalf(Eigenvals(A2,X));\nLambda := diag(seq(eigenvalues
[k],k=1..rowdim(A2)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,eigenvalu
esG-%'vectorG6#7$$\"+&z_'\\K!\")$\"+@Z.X5!\"(" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%'LambdaG-%'matrixG6#7$7$$\"+&z_'\\K!\")\"\"!7$F-$\"+@
Z.X5!\"(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalm(X);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$$!+C\\0!))*!#6$\"+#o\"
\\C6!#57$$!+%Q(=W:F-$!+I\\*QO)F-" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 31 "evalm( A2 &* X - X &* Lambda );" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'matrixG6#7$7$$\"\"\"!\"*\"\"!7$$!\"$F*$\"\"&!\")" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "det(X);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"+)*********!#6" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "charpoly(A2, lambda);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,(*$)%'lambdaG\"\"#\"\"\"\"\"\"F&!$P\"\"%'R$F)" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 12 "fsolve( % );" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$$\"+&z_'\\K!\")$\"+@Z.X5!\"(" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 43 "det( A2 - lambda * matrix(2,2,[1,0,0,1]) );" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%'lambdaG\"\"#\"\"\"\"\"\"F&!$P\"
\"%'R$F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "A := evalm(transpose(A2) &* A2); # \+
symmetric  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7$7$
\"&u1\"!%;*)7$F+\"%G&)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "e
igenvalues := evalf(Eigenvals(A,Q));" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%,eigenvaluesG-%'vectorG6#7$$!+9.%H+\"!\"%$!+J0!*\\6!\"'" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalm(Q);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#-%'matrixG6#7$7$$!+C\\0!))*!#6$\"+#o\"\\C6!#57$$!+%Q(
=W:F-$!+I\\*QO)F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
{MARK "26 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }