4.
Orthogonal transformations and Rotations
A matrix is defined to be orthogonal if the entries are real and
(1)
A
0
A
=
I
.
Condition (1) says that the gram matrix of the sequence of vectors formed by the
columns of
A
is the identity, so the columns are an orthonormal frame. An orthog
onal matrix defines an orthogonal transformation by mutiplying column vectors on
the left.
Condition (1) also shows that
A
is a rigid motion preserving angles and distances.
A matrix satisfying (1) preserves the dot product; the dot product of two vectors
is the same before and after an orthogonal transformation. This can be written as
Av
·
Aw
=
v
·
w
for all vectors
v
and
w
. This is true by the definition (1) of orthogonal matrix since
Av
·
Aw
= (
Av
)
0
Aw
=
v
0
A
0
Aw
=
v
0
Iw
=
v
0
w
=
v
·
w
.
Thus lengths and angles are preserved, since they can be written in terms of dot
products.
The orthogonal transformation are a
group
since we can multiply two of them
and get an orthogonal transformation. This is because if
A
and
B
are orthogonal,
then
A
0
A
=
I
and
B
0
B
=
I
. So
(
AB
)
0
AB
=
B
0
A
0
AB
=
I
,
showing that
AB
is also orthogonal. Likewise we can take the inverse of an orthog
onal transformation to get an orthogonal transformation.
Orthogonal transformations have determinant 1 or

1 since by (1) and properties
of determinant,
(det
A
)
2
= det(
A
0
) det
A
= det(
A
0
A
)
= det
I
= 1
.
4.1.
The rotation group.
Orthogonal transformations with determinant 1 are
called rotations, since they have a fixed axis.
This is discussed in more detail
below. The rotations also form a group.
If we think of an orthogonal matrix
A
as a frame
A
= (
v
1
,
v
2
,
v
3
)
,
then the determinant is the scalar triple product
v
1
·
(
v
2
×
v
3
)
.
The frame is right handed if the triple product is 1 and left handed if it is 1. The
frame is the image of the right handed standard frame
(
e
1
,
e
2
,
e
3
) =
I
under the transformation
A
.
Thus
A
preserves orientation (righthandedness) if
the determinant is 1.
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
2
4.1.1.
Rotations and cross products.
Rotations are orthogonal transformations which
preserve orientation.
This is equivalent to the fact that they preserve the vector
cross product:
(2)
A
(
v
×
w
) =
Av
×
Aw
,
for all rotations
A
and vectors
v
and
w
. Recall the right hand rule in the definition
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '11
 staff
 Dot Product, Rotation, Rotation matrix, Rz

Click to edit the document details