LINEAR ALGEBRA
W W L CHEN
c
W W L Chen, 1997, 2005.
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Chapter 10
ORTHOGONAL MATRICES
10.1. Introduction
Definition.
A square matrix
A
with real entries and satisfying the condition
A
−
1
=
A
t
is called an
orthogonal matrix.
Example 10.1.1.
Consider the euclidean space
R
2
with the euclidean inner product.
The vectors
u
1
= (1
,
0) and
u
2
= (0
,
1) form an orthonormal basis
B
=
{
u
1
,
u
2
}
. Let us now rotate
u
1
and
u
2
anticlockwise by an angle
θ
to obtain
v
1
= (cos
θ,
sin
θ
) and
v
2
= (
−
sin
θ,
cos
θ
). Then
C
=
{
v
1
,
v
2
}
is
also an orthonormal basis.
θ
u
1
u
2
v
1
v
2
Chapter 10 : Orthogonal Matrices
page 1 of 10
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Linear Algebra
c
W W L Chen, 1997, 2005
The transition matrix from the basis
C
to the basis
B
is given by
P
= ([
v
1
]
B
[
v
2
]
B
) =
cos
θ
−
sin
θ
sin
θ
cos
θ
.
Clearly
P
−
1
=
P
t
=
cos
θ
sin
θ
−
sin
θ
cos
θ
.
In fact, our example is a special case of the following general result.
PROPOSITION 10A.
Suppose that
B
=
{
u
1
, . . . ,
u
n
}
and
C
=
{
v
1
, . . . ,
v
n
}
are two orthonormal
bases of a real inner product space
V
. Then the transition matrix
P
from the basis
C
to the basis
B
is
an orthogonal matrix.
Example 10.1.2.
The matrix
A
=
1
/
3
−
2
/
3
2
/
3
2
/
3
−
1
/
3
−
2
/
3
2
/
3
2
/
3
1
/
3
is orthogonal, since
A
t
A
=
1
/
3
2
/
3
2
/
3
−
2
/
3
−
1
/
3
2
/
3
2
/
3
−
2
/
3
1
/
3
1
/
3
−
2
/
3
2
/
3
2
/
3
−
1
/
3
−
2
/
3
2
/
3
2
/
3
1
/
3
=
1
0
0
0
1
0
0
0
1
.
Note also that the row vectors of
A
, namely (1
/
3
,
−
2
/
3
,
2
/
3), (2
/
3
,
−
1
/
3
,
−
2
/
3) and (2
/
3
,
2
/
3
,
1
/
3) are
orthonormal. So are the column vectors of
A
.
In fact, our last observation is not a coincidence.
PROPOSITION 10B.
Suppose that
A
is an
n
×
n
matrix with real entries. Then
(a)
A
is orthogonal if and only if the row vectors of
A
form an orthonormal basis of
R
n
under the
euclidean inner product; and
(b)
A
is orthogonal if and only if the column vectors of
A
form an orthonormal basis of
R
n
under the
euclidean inner product.
Proof.
We shall only prove (a), since the proof of (b) is almost identical. Let
r
1
, . . . ,
r
n
denote the row
vectors of
A
. Then
AA
t
=
r
1
·
r
1
. . .
r
1
·
r
n
.
.
.
.
.
.
r
n
·
r
1
. . .
r
n
·
r
n
.
It follows that
AA
t
=
I
if and only if for every
i, j
= 1
, . . . , n
, we have
r
i
·
r
j
=
1
if
i
=
j
,
0
if
i
=
j
,
if and only if
r
1
, . . . ,
r
n
are orthonormal.
PROPOSITION 10C.
Suppose that
A
is an
n
×
n
matrix with real entries. Suppose further that the
inner product in
R
n
is the euclidean inner product. Then the following are equivalent:
(a)
A
is orthogonal.
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