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Applications
of Quadratic
Forms

8
...
This sec­
tion may be omitted with no loss of continuity
...

However, a theorem such as the Principal Axis Theorem is often important because
it provides a simple way of thinking about complicated situations
...


Small Deformations
A small deformation of a solid body may be understood as the composition of three
stretches along the principal axes of a symmetric matrix together with a rigid rotation
of the body
...
This might be, for example, a piece of steel under some load
...
Suppose that a material point in the body,
which is at 1 before the forces are applied, is moved by the forces to the point f (1) =
(f1(1), f2(1), f3(1)); we have assumed that f(O)
0
...
(Note that f

represents the displacement of the point initially at

1, not the force at 1
...


It is convenient to introduce a parameter f3 to

describe how small the deformation is and a function

f(x) = 1

+

h(x) and write

f3h(x)

h(x) in terms of the point x, the given function

This equation is really the definition of

f(x), and the parameter {3
...
IR
...
In this case, the map

f

approximated near the origin by the linear transformation with matrix

:

�

(0)

is

so that

[�;� (0) J v
...
)
In terms of the parameter f3 and the function

[�CO)]=

I+

h, this matrix can be written as

f3G

= [��� (0)J
...
2
...

The next step is to observe that we can write

I + = I + + = (I +
f3G

f3(E

W)

{3E)(J + f3W) - /32 EW

Since f3 is assumed to be small, {32 is very small and may be ignored
...
)

The small deformation we started with is now described as the composition of two
linear transformations, one with matrix

f3E and the other with matrix

I+

f3W
...
(The matrix f3W is called an
infinitesimal rotation
...
This matrix is sym­
be shown that

I+

I+

metric, so there exist principal axes such that the symmetric matrix is diagonalized to

[ � 1 � � ]·
1+€3
I
Ei

l

E2

0

because

(It is equivalent to diagonalize

f3E

and add the result to

0

I,

is transformed to itself under any orthonormal change of coordinates
...
This diagonalized matrix can be written as the product of the three matrices:

l3 = [1+ o1 ol [1 1+
1+ €
1
0
0

0

fl

0
0

0

0

0
0

f2

0

It is now apparent that, excluding rotation, the small deformation can be represented
as the composition of three stretches along the principal axes of the matrix
quantities

E1, E2,

f3E
...
(/3£ is called the infinitesimal strain; this notation
is not quite the standard notation
...
)

The Inertia Tensor
For the purpose of discussing the rotation motion of a rigid body, information
about the mass distribution within the body is summarized in a symmetric matrix N
called the inertia tensor
...


(2) In general,

the angular momentum vector

where

l of the rotating body is equal to Nw,

instantaneous angular velocity vector
...
This is a beginning to an explanation of how the body wob­
bles during rotation

(w

need not be constant) even though

l is

a conserved quantity

(that is, /is constant if no external force is applied)
...
Make this fixed point the origin (0, 0, 0)
...
At any time

have moved to a new position
...
3

is deter­

u(t + 6-t) and denote
the angle by 6-8
...
Also, as M
0,
��
'f*, the instantaneous rate of rotation about the axis
...


mined by its axis and an angle
...
)

To use concepts such as energy and momentum in the discussion of rotating mo­

tion, it is necessary to introduce moments of inertia
...
The factor (XT + x�)

is simply the square of the distance of the mass from the x3-axis
...


For a general axis e through the origin with unit direction vector it, the moment of
inertia of the mass about e is defined to be
of

m from e
...


m multiplied

the moment of inertia in this case is

u)uf [1 - (1
...
u)2)

With some manipulation, using ar i1 = 1 and
equal to the expression

Because of this, for the single point mass

the 3 x 3 matrix

by the square of the distance

1 i1 = 1T i1,

m at 1,

·

we can verify that this is

we define the inertia tensor N to be

(Vectors and matrices are special kinds of "tensors"; for our present purposes, we
simply treat N as a matrix
...
It is clear that this matrix N is symmetric because

xxT is a symmetric 3 x 3 matrix
...
This

name has no special meaning; the term is simply a product that appears as an entry in
the inertia tensor
...
Consider a rigid body
that can be thought of as k masses joined to each other by weightless rigid rods
...
The
inertia tensor of this body is just the sum of the inertia tensors of the k masses; since
it is the sum of symmetric matrices, it is also symmetric
...
In any case, the inertia tensor N is still defined, and is still a symmetric
matrix
...
The diagonal entries are then the moments of inertia with respect
to the principal axes, and these are called the principal moments of inertia
...
Suppose
an arbitrary axis t is determined by the unit vector ii such that [il]p

=

Then,

from the discussion of quadratic forms in Section 8
...

It is important to get equations for rotating motion that corresponds to Newton's
equation:
The rate of change of momentum equals the applied force
...

It turns out that the right way to define the angular momentum vector
body is

J'

=

l for

a general

N(t)w(t)

Note that in general N is a function oft since it depends on the positions at time t of
each of the masses making up the solid body
...
In

general, this is a very difficult problem, but there will often be important simplifications

N is diagonalized by the Principal Axis Theorem
...


if

PROBLEM 8
...


P also

CHAPTER REVIEW
Suggestions for Student Review
1

How does the theory of diagonalization of symme­
tric matrices differ from the theory for general square
matrices? (Section 8
...
How do you find the symmetric
matrix corresponding to a quadratic form? How does
diagonalization of the symmetric matrix enable us to
diagonalize the quadratic form? (Section

8
...
How
does

diagonalizing

the

corresponding

symme­

tric matrix help us classify a quadratic form?
(Section

4 What role do eigenvectors play in helping us under­
stand the graphs of equations Q(x)= k, where Q(x)

8
...

How do the principal axes of A relate to the graph of
Q(x) =
...
3)
6 When diagonalizing a symmetric matrix A, we know
that we can choose the eigenvalues in any order
...
(Section

8
...
3)

Chapter Quiz
El Let A =

[-� � �]·
-

2

E3 By diagonalizing the quadratic form, make a sketch
Find an orthogonal matrix

3 2

P such that pT AP=

D is diagonal
...

(ii) Express Q(x) in diagonal form and give the or­
matrix

thogonal matrix that brings it into this form
...


(iv) Describe the shape of

in the x1x2-plane
...


E4 Prove that if A is a positive definite symmetric ma­
trix, then (1, y) = _xT Ay is an inner product on JR11•
ES Prove that if A is a 4

Q(x) = 1 and Q(x)= 0
...


= (tl

- 3)4,

then

Further Problems
Fl In Problem 7
...
Let Ai

=

RQ, and prove that

F4 (a) Suppose that A is an invertible n x n matrix
...
This is

Ai is orthogonally similar to A and hence has the

known as a polar decomposition of A
...
(By repeating this process,

Use Problems F2 and F3, let U be the square
root of ATA, and let Q
AU 1 )

A

=

QiR1, Ai

=

Ri Qi, A1

=

QzR 2, Az

=

RzQ 2,
...
)

symmetric matrix
...

That is, show that there is a positive semidefinite
symmetric matrix B such that 82
A
...


Define C to be a positive square root for D and let

B

=

QCQT
...
(Hint: Con­
sider Ax· Ax
...


(b) Let V
that A

F2 Suppose that A is an n x n positive semidefinite

pose that Q diagonalizes A to D so that QTAQ

-

=

=

=


...
Show that Vis symmetric and
VQ
...

(c) Suppose that the 3 x 3 matrix A is the matrix
of an orientation-preserving linear mapping L
...
(This follows from part (a), facts
about isometries of JR3, and ideas in Section

8
...
In fact, this is a finite version of the result
for infinitesimal strain in Section 8
...
)



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