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{SECT 0 {EXCHG {PARA 260 "" 0 "" {TEXT 256 2 "\011\011" }{TEXT 257 32 
"\011Quadratic Forms and Eigenvalues" }}}{EXCHG {PARA 256 "" 0 "" 
{TEXT 258 368 "This worksheet explores the relationship between symmet
ric matrices and\nquadratic forms. It shows how the eigenvalues of suc
h a matrix relate to\nthe geometric character of the graph of the quad
ratic form. It also discusses\nin the context of an example how the ei
genvectors of the symmetric matrix\ndetermine a rotation of coordinate
s making the quadratic form diagonal." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 29 "with(plots): with(plottools):" }}{PARA 7 "" 1 "" 
{TEXT -1 50 "Warning, the name changecoords has been redefined\n" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}{PARA 7 "" 
1 "" {TEXT -1 80 "Warning, the protected names norm and trace have bee
n redefined and unprotected\n" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 259 
57 "The symmetric matrix corresponding to 2 x^2 + 8 x y + y^2" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "A := matrix(2,2,[2,4,4,1]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7$7$\"\"#\"\"%7$F+\"
\"\"" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 260 24 "A general vector in R^
2." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "v := vector(2,[x,y]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG-%'vectorG6#7$%\"xG%\"yG" }}}
{EXCHG {PARA 256 "" 0 "" {TEXT 261 58 "Verifying that A corresponds to
 this quadratic expression." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "expn
1 := expand(dotprod(v,evalm(A &* v)));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%&expn1G,**&%\"xG\"\"\"-%*conjugateG6#F'F(\"\"#*(\"\"%F(F'F(-F*
6#%\"yGF(F(*(F.F(F1F(F)F(F(*&F1F(F/F(F(" }}}{EXCHG {PARA 256 "" 0 "" 
{TEXT 262 199 "Compute the eigenvalues (returned as a list below) and \+
eigenvectors\n(returned  in the columns of A_evects) of the matrix A.
\nRecall a nonzero v is eigenvector with eigenvalue lambda if A v = la
mbda v." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "evalf(Eigenvals(A,A_evec
ts));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7$$!+s)G6`#!\"*$
\"+w)G6`&F)" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 263 52 "The columns her
e are the corresponding eigenvectors." }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 16 "print(A_evects);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#
7$7$$\"+Mc-=m!#5$!+d<y'\\(F*7$F+$!+Mc-=mF*" }}}{EXCHG {PARA 256 "" 0 "
" {TEXT 264 103 "Check that the columns of A_evects have been returned
 as unit vectors.\nIf not, we'd normalize them now." }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 70 "for j from 1 to 2 do\n\011print(sqrt(sum((A_evects
[i,j])^2,i=1..2)));\n\011od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++
++5!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++++5!\"*" }}}{EXCHG 
{PARA 256 "" 0 "" {TEXT 265 93 "(printf(`Length of column %d is %g\\n`
,j,sqrt(sum((A_evects[i,j])^2,i=1..2))) would be nicer.)" }}{PARA 256 
"" 0 "" {TEXT 266 206 "The eigenvectors of a symmetric matrix can alwa
ys be chosen to be an orthonormal\nbasis for R^n, and this reflects it
self in the identity below.\nThis identity also means A_evects is the \+
matrix of a rotation." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "print(eval
m(inverse(A_evects) - transpose(A_evects)));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'matrixG6#7$7$$\"\"!F)F(F'" }}}{EXCHG {PARA 256 "" 0 
"" {TEXT 267 308 "In general, if an n x n  matrix A has n independent \+
eigenvectors in the columns of\nA, then  inverse (A_evects) &* A &* A_
evects will be a diagonal matrix.\n\nIf we transform coordinates by [x
_new y_new] = A_evects &* [x y], then B will\nbe the symmetric matrix \+
corresponding to expn1 in coordinates x_new,y_new." }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 49 "B := evalm(transpose(A_evects) &* A &* A_evects);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'matrixG6#7$7$$!+v)G6`#!\"*$
!\"\"F,7$F-$\"+u)G6`&F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "
v_new := vector(2,[x_new,y_new]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%&v_newG-%'vectorG6#7$%&x_newG%&y_newG" }}}{EXCHG {PARA 256 "" 0 "" 
{TEXT 268 43 "The same expression in the new coordinates." }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 50 "expn2 := expand(dotprod(v_new,evalm(B &* v_n
ew)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&expn2G,**&%&x_newG\"\"\"-
%*conjugateG6#F'F($!+v)G6`#!\"**($F(F.F(F'F(-F*6#%&y_newGF(!\"\"*($F(F
.F(F3F(F)F(F4*($\"+u)G6`&F.F(F3F(F1F(F(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 67 "plot3d(expn1,x=-5..5,y=-5..5,axes=boxed,shading=zhue,
view=-10..10);" }}{PARA 13 "" 1 "" {GLPLOT3D 308 308 308 {PLOTDATA 3 "
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