This preview shows page 1. Sign up to view the full content.
3.9. THE PRINCIPAL AXIS THEOREM
375
What does this result tell us about quadratic forms?
We start with some consequences of the second property of the
eigenvectors
−→
u
i
:
−→
u
i
·
−→
u
j
=
b
0 if
i
n
=
j,
1 if
i
=
j.
Geometrically, this says two things: Frst (using
i
=
j
) they are
unit
vectors
(

−→
u
i

2
−
−→
u
i
·
−→
u
i
= 1), and second, they are mutually perpendicular. A
collection of vectors with both properties is called an
orthonormal
set of
vectors. Since they deFne three mutually perpendicular directions in space,
a set of three orthonormal vectors in
R
3
can be used to set up a new
rectangular coordinate system: any vector
−→
x
∈
R
3
can be located by its
projections onto the directions of these vectors, which are given as the dot
products of
−→
x
with each of the
−→
u
i
. You should check that using these
coordinates, we can express any vector
−→
x
∈
R
3
as a linear combination of
the
−→
u
i
:
−→
x
= (
−→
x
·
−→
u
1
)
−→
u
1
+ (
−→
x
·
−→
u
2
)
−→
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Spring '08 term at University of Florida.
 Spring '08
 ALL
 Calculus, Eigenvectors, Vectors

Click to edit the document details