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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 56 "6.5 Quadratic Forms; Ort
hogonal Diagonalization - Part 2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 30 "In-class demo by Russell Blyth" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "restart: with(linalg): with(
plots): with(plottools):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 42 "Example 3: The quadric surface 2xy + z = \+
0" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "C is
 the matrix of the quadratic form" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 44 "C := matrix(3,3,[[0,1,0],[1,0,0],[0,0,0]]);\n" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "Find the eigenvalues and eigenvect
ors of B" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "evC := [eigenve
ctors(C)];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 27 "Normalize the eigenvectors:" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 92 "u1 := normalize(evC[1][3][1]);\nu2 := normali
ze(evC[2][3][1]);\nu3 := normalize(evC[3][3][1]);" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 54 "Construct a matrix P which orthogonally diagonaliz
es A" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "PC := augment(u1,u2
,u3);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "Compute the diagonal mat
rix (since A = PD" }{XPPEDIT 18 0 "P^t" "6#)%\"PG%\"tG" }{TEXT -1 14 "
, we have D = " }{XPPEDIT 18 0 "P^t" "6#)%\"PG%\"tG" }{TEXT -1 3 "AP)
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "DC := evalm(transpose(P
C) &* C &* PC);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "Make Maple com
pute the equation of the rotated conic:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 96 "LHSC := evalm(transpose([X,Y,Z]) &* DC &* [X,Y,Z]) + \+
evalm(transpose([0,0,1]) &* PC &* [X,Y,Z]);" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "The equati
on is LHSC = 0, which is the equation of a hyperboloid of one sheet." 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "Let's g
raph, first the quadric surface relative to the new axes:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "implicitplot3d(x^2-z^2+y=0,x=-10..
10,y=-10..10,z=-10..10,axes=normal,scaling=constrained,grid=[20,20,20]
);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 130 "And then relative to the o
riginal axes: (the eigenspace basis vectors have been stretched to mak
e them long enough to see clearly)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 59 "u15 := evalm(5*u1); u25 := evalm(5*u2); u35 := evalm(
5*u3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 336 "u1g := line([0,0
,0],convert(u15,list),color=red,thickness=4):\nu2g := line([0,0,0],con
vert(u25,list),color=red,thickness=4):\nu3g := line([0,0,0],convert(u3
5,list),color=red,thickness=4):\nconicB := implicitplot3d(2*x*y + z = \+
0, x=-10..10,y=-10..10,z=-10..10,axes=normal,grid=[20,20,20]):\ndispla
y([u1g,u2g,u3g,conicB],scaling=constrained);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "1 0 0" 0 }{VIEWOPTS 1 1 0 3 2 1804 
1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }