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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 24 "Moment of Inertia Tensor
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 200 "In t
his worksheet we calculate the moment of inertia matrix in a given coo
rdinate system and show that a coordinate system can be found in which
 the inertia tensor is represented by a diagonal matrix." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "This worksheet req
uires " }{TEXT 19 6 "maple8" }{TEXT -1 178 " to work properly (integra
l I2 below won't evaluate in earlier versions, and even in maple8 it i
ntegrates with a wrong sign). For a workaround that involves some extr
as: consult " }{TEXT 19 18 "MomentInertia7.mws" }{TEXT -1 20 " in the \+
same folder." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 29 "restart; with(LinearAlgebra):" }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 29 "with(plots): with(plottools):" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 237 "We start with an example for which the calculatio
n of the integrals is relatively straightforward. We assume a homogene
ous material for which the mass density is constant, and given as the \+
ratio of total mass divided by the total volume." }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 9 "rho:=M/V;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
173 "To practice our computational skills we work on an ellipsoid whic
h is described by a quadratic form. This will be a cigar-shaped body, \+
which has an ellipse as cross section." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 45 "A:=Matrix([[1,1/2,1/3],[1/2,4,1],[1/3,1,4]]);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "A-Transpose(A);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 71 "The above null-result shows that A was ch
osen to be real and symmetric." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 29 "EVals:=evalf(Eigenvalues(A));" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 207 "The numerical noise in the eigenvalues (non-zero imagina
ry parts) comes from the evalf calculation to 10 digits (verify this b
y repeating the calculation with Digits set higher than the default va
lue of 10)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 196 "The matrix has positive eigenvalues which are related to the s
emimajor axes of the ellipsoid. If one (or two) of them were negative \+
our quadratic surface would be made up of hyperboloidal sheets. " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "We antic
ipate a result to be derived below: in a principal-axis frame, the qua
dratic form will be given as:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 19 47 "QF = EVals[1]*x^2 + EVals[2]*y^2 + EVals[
3]*z^2" }}{PARA 0 "" 0 "" {TEXT -1 136 "We can set QF =1 (to chose a s
cale), and recognize that the eigenvalues are equal to EV[i] = 1/a_i^2
, where a_i are the semi-major axes." }}{PARA 0 "" 0 "" {TEXT -1 55 "T
herefore the volume of the ellipsoid is calculated as:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "V0:=evalf(4/3*Pi*mul(sqrt(1/EVals[i
]),i=1..3));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "seq(Re(sqrt
(1/EVals[i])),i=1..3);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Now we \+
calculate the quadratic form." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 20 "Rv:=Vector([x,y,z]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
27 "QF:=Transpose(Rv) . A . Rv;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 17 "QF:=simplify(QF);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
20 "surf:=solve(QF-1,z);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
116 "plot3d(surf[1],x=-1.1..1.1,y=-1.1..1.1,axes=boxed,grid=[30,30],st
yle=patchcontour,shading=zhue,scaling=constrained);" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 67 "This doesn't look so great, so it is better to \+
use an implict plot." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "im
plicitplot3d(QF-1,x=-1.1..1.1,y=-1.1..1.1,z=-1.1..1.1,axes=boxed,numpo
ints=2000,scaling=constrained,style=patchcontour,shading=zhue);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "EVecs:=evalf( Eigenvectors(A
)  );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "We are interested in a \+
graph of the three eigenvectors. They are stored in the columns of the
 matrix (2nd entry in " }{TEXT 19 5 "Evecs" }{TEXT -1 108 "). We want \+
to show that they are orthogonal, and how they are related to the shap
e of the quadratic surface." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
51 "E1v:=convert(SubMatrix(EVecs[2],1..3,1..1),Vector);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "E1v:=Normalize(map(Re,E1v),Euclidea
n);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "E2v:=Normalize(map(R
e,convert(SubMatrix(EVecs[2],1..3,2..2),Vector)),Euclidean);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "E3v:=Normalize(map(Re,conver
t(SubMatrix(EVecs[2],1..3,3..3),Vector)),Euclidean);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 62 "E1 := plottools[arrow]([0, 0, 0], E1v, .2
, .4, .1, color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "E2
 := plottools[arrow]([0, 0, 0], E2v, .2, .4, .1, color=blue):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "E3 := plottools[arrow]([0, 0
, 0], E3v, .2, .4, .1, color=green):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 135 "QFp:=implicitplot3d(QF-1,x=-1.1..1.1,y=-1.1..1.1,z=-
1.1..1.1,axes=boxed,numpoints=2000,scaling=constrained,style=wireframe
,color=gold):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "display(E1
,E2,E3,QFp,axes=boxed);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "
E1v . E2v;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "E1v . E1v;" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "E1v . E3v;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "E2v . E3v;" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 71 "Rotate the above graph to obtain a clear understanding
 of the geometry." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT 259 11 "Exercise 1:" }}{PARA 0 "" 0 "" {TEXT -1 67 "Change the m
atrix in the above quadratic form and repeat the steps." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 177 "We proceed \+
with constructing an orthogonal matrix out of the normalized eigenvect
ors. We then transform the matrix to demonstrate that it is diagonaliz
ed by this transformation." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
25 "R:=Matrix([E1v,E2v,E3v]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 21 "Ri:=MatrixInverse(R);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 124 "O
ur transformation matrix made up of the eigenvectors as columns is ort
hogonal, i.e., the inverse is given by the transpose." }}{PARA 0 "" 0 
"" {TEXT -1 148 "We now calculate the transformation of matrix A by th
e matrix R, the correct form is Ri . A . R  (the other way around does
n't work as shown below!)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
11 "Ri . A . R;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 116 "The transform
ed matrix has the eigenvalues on the diagonal, and zeroes (approximate
ly) for all non-diagonal entries." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 11 "R . A . Ri;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 260 11 "
Exercise 2:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 226 "Calculate the quadratic form using the diagonalized matrix. Tr
uncate the near-zero entries to zero, i.e., make up a diagonal matrix \+
from the eigenvalues and observe the equation for the quadratic form (
absence of cross terms)." }}{PARA 0 "" 0 "" {TEXT -1 129 "Carry out th
e transformation to the principal axes frame (as done above) for some \+
other matrix A with three positive eigenvalues." }}{PARA 0 "" 0 "" 
{TEXT -1 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 261 19 "Volume calculat
ion." }}{PARA 0 "" 0 "" {TEXT -1 28 "Can we calculate the volume " }
{TEXT 258 1 "V" }{TEXT -1 76 " in Cartesian coordinates? We choose the
 quadratic form with scale factor 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 58 "Let us do the repeated integrals one at
 a time. First the " }{TEXT 257 1 "z" }{TEXT -1 13 " integration:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "fz:=solve(QF-1,z);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "I1:=int(1,z=fz[1]..fz[2]);" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "fy:=solve(I1,y);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "plot([fy[1],fy[2]],x=-1.1..1
.1,color=[red,blue],numpoints=800,scaling=constrained);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 49 "Following the z-integration we need to in
tegrate " }{TEXT 19 2 "I1" }{TEXT -1 36 " over an area bounded by the \+
curves " }{TEXT 19 5 "fy[1]" }{TEXT -1 5 " and " }{TEXT 19 5 "fy[2]" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 157 "We will integrate over \+
y between the lower (blue) and upper (red) bounding curves. Then we wi
ll integrate over x between fixed limits. What are these limits?" }}
{PARA 0 "" 0 "" {TEXT -1 63 "They are the points where the derivative \+
y(x) goes to infinity." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "x
0:=solve(1/diff(fy[1],x),x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "T
he direct attempt of definite integration is not successful:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "I2:=int(I1,y=fy[1]..fy[2]);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "I3idf:=int(expand(I2),x
);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "We should be able to do thi
s integral at least numerically. We need to know the range of x:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "evalf(x0);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 24 "plot(I2,x=x0[1]..x0[2]);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 190 "Somehow maple8 made a mistake: it chose \+
a branch cut for the ln function which turned our results upside down.
 We need to integrate this function, and do it numerically and correct
 the sign:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "V:=evalf(Int(
-I2,x=x0[1]..x0[2]));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "V0
:=evalf(4/3*Pi*mul(sqrt(1/Eigenvalues(A)[i]),i=1..3),15);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 125 "The volume integration worked! We carrie
d out two of the three integrations analytically, and then the third o
ne numerically." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 21 "Now we calculate the " }{TEXT 264 25 "moments of inertia \+
matrix" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "M:=1; rho;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "MI:=Matrix(3,3):" }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 43 "for i from 1 to 3 do: for j from 1 to i do:" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 149 "if i=1 and j=1 then w:=Rv[2]^2+Rv[
3]^2; elif i=2 and j=2 then w:=Rv[1]^2+Rv[3]^2; elif i=3 and j=3 then \+
w:=Rv[1]^2+Rv[2]^2; else w:=-Rv[i]*Rv[j]; fi:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 30 "I1:=int(w*rho,z=fz[1]..fz[2]);" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 28 "I2idf:=unapply(int(I1,y),y);" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 38 "I2:=expand(I2idf(fy[2])-I2idf(fy[1]));" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 17 "I2:=simplify(I2);" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 39 "MI[i,j]:=evalf(Int(I2,x=x0[1]..x0[2]));" }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 7 "od: od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 10 "print(MI);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "Now fill th
e complete matrix by symmetry:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 69 "for i from 1 to 3 do: for j from i to 3 do: MI[i,j]:=MI[j,i]: \+
od: od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "print(MI);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "Eigenvalues(MI);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 43 "These are the principal moments of inerti
a." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "EVecsMI:=Eigenvectors
(MI);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "We are interested in a g
raph of the three eigenvectors." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 55 "E1vMI:=convert(SubMatrix(EVecsMI[2],1..3,1..1),Vector);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "E1vMI:=Normalize(map(Re,E1vM
I),Euclidean);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "E2vMI:=No
rmalize(map(Re,convert(SubMatrix(EVecsMI[2],1..3,2..2),Vector)),Euclid
ean);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "E3vMI:=Normalize(m
ap(Re,convert(SubMatrix(EVecsMI[2],1..3,3..3),Vector)),Euclidean);" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "RMI:=Matrix([E1vMI,E2vMI,E3
vMI]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "R;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "E1MI := plottools[arrow]([0, 0, 0],
 E1vMI, .2, .4, .1, color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 67 "E2MI := plottools[arrow]([0, 0, 0], E2vMI, .2, .4, .1, color=b
lue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "E3MI := plottools[
arrow]([0, 0, 0], E3vMI, .2, .4, .1, color=green):" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 124 "The eigenvectors of the MI matrix provide those \+
axes about which rotation of the body without deviation moments is pos
sible." }}{PARA 0 "" 0 "" {TEXT -1 254 "Note that as in the 'spinning \+
textbook' problem the motion about the axes with extreme eigenvalues w
ill be truly stable, while free rotation about the axis associated wit
h the intermediate eigenvalue results in a wobble (instability in Eule
r's equation)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 411 "The remarkable thing about rotational mechanics is that \+
for any shape (e.g., a potato) three axes can be found about which pur
e rotation is possible without deviation moments. The ellipsoid is as \+
close a mathematical form that we can handle that describes such a pot
ato. The diagram below illustrates that the stable rotation axes do co
incide with the axes that are perpendicular to the elliptic cross sect
ions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 39 "display(E1MI,E2MI,E3MI,QFp,axes=boxed);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 221 "The eigenvectors of the moment of inerti
a matrix look the same as the eigenvectors that diagonalize the quadra
tic form. Here is an example of two matrices that share a set of eigen
vectors, but have different eigenvalues. " }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 385 "We observe that vectors can be fo
und which are at the same time eigenvectors of matrix A (which defines
 the geometric shape), and of the moment of inertia matrix MI. These t
wo matrices are distinct and have distinct eigenvalues. According to a
 theorem in linear algebra a common set of eigenvectors can be found f
or two matrices if they commute with each other, i.e., wenn the produc
t " }{TEXT 19 3 "A.B" }{TEXT -1 16 " is the same as " }{TEXT 19 3 "B.A
" }{TEXT -1 2 " :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 30 "Let us explore the properties:" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "First, the orthogonality of the
 eigenvectors of the MI matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 14 "E1vMI . E2vMI;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "E
1vMI .E1vMI;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 106 "Of course, the e
igenvectors of the MI matrix are perpendicular. Try the others by edit
ing the above lines." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 121 "The following three lines indicate which eigenvectors \+
of the MI matrix correspond to the principal axes of the ellipsoid:" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "E1v . E1vMI;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "E2v . E1vMI;" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 12 "E3v . E1vMI;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 99 "The first eigenvector of the MI matrix corresponds to the secon
d eigenvector of the quadratic form." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "E1v . E2vMI;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 12 "E2v . E2vMI;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "E3v .
 E2vMI;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 99 "The second eigenvector
 of the MI matrix corresponds to the first eigenvector of the quadrati
c form." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "E1v . E3vMI;" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "E2v . E3vMI;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "E3v . E3vMI;" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 120 "Here we have a sign change. Eigenvectors are define
d up to a multiplicative constant. We display all vectors and teh QF:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "display(E1,E2,E3,E1MI,E
2MI,E3MI,QFp,axes=boxed);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 102 "Let
 us calculate the commutator of the two matrices, namely the one defin
ing the QF and the MI matrix:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 16 "A . MI - MI . A;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "Indeed,
 the product " }{TEXT 19 6 "A . MI" }{TEXT -1 54 " is within numerical
 accuracy the same as the product " }{TEXT 19 6 "MI . A" }{TEXT -1 2 "
 !" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 263 11 "Ex
ercise 3:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
144 "Explore the relationship between the principal axes of the quadra
tic form, and the principal axes of the moment of inertia matrix. Try \+
matrices " }{TEXT 19 1 "A" }{TEXT -1 85 " for which the form is close \+
to a nearly circular cross section, i.e., find matrices " }{TEXT 19 1 
"A" }{TEXT -1 50 " such that two of the eigenvalues are very close. " 
}}{PARA 0 "" 0 "" {TEXT -1 125 "What happens to the eigenvectors when \+
two eigenvalues are very close? What shape do you get when three eigen
values are close?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "104" 0 }{VIEWOPTS 1 1 0 3 2 
1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }