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Orthonormal Bases and
Orthogonal Matrices

7
...
11• It is therefore a little uncomfortable for many beginners to deal
with the arbitrary bases that arise in Chapter 4
...


Orthonormal Bases
Definition

A set of vectors {V1, ••• vk} in JRll is orthogonal if vi· v1
,

=

O whenever i

*

j
...
) The set

is an octhogonal set of vectors in 11!
...


If the zero vector is excluded, orthogonal sets have one very nice property
...


•

, vd is an orthogonal set of non-zero vectors in JR'Z, it is linearly

independent
...


Take the dot product of v; with

each side to get

Cc1v1 +
...
v;) +
...
v;) +
...
v;)
0 +
...
+ 0
since v; · v1

=

0 unless i

=

j
...


0,

=

O·v;

=

o

=

0

so llv;ll f
...
Since

this is true for all 1 � i � k, it follows that { v 1,
...


...
Many of the things we do with orthogonal sets
depend on it
...


Definition

A set {V1,

Orthonormal

is a unit vector (that is, each vector is normalized)
...
It follows from Theorem 1 that orthonormal sets are necessarily
linearly independent
...
For example, in

JR6' {e,' e2, es, e6} is an orthonormal set of four vectors (where, as usual, e; is the i-th
standard basis vector)
...
The vectors are multipies

of the vectors in Example 1, so they are certainly mutually orthogonal
...


EXERCISE 1
Verify that the set

{ [il r�l [-�]}
·

·

is orthogonal and then normalize the vectors to

produce the corresponding orthonormal set
...
However, the general arguments are slightly simpler for orthonor­
mal sets since llvill

=

1 in this case
...
(Compare Examples 1 and 3
...


Coordinates with Respect to an Orthonormal Basis
An orthonormal set of n vectors in JR
...
12 since it is auto­
matically linearly independent and a set of n linearly independent vectors in JR
...
11 by Theorem 4
...
4
...

The first of these advantages is that it is very easy to find the coordinates of a vector

{v1,
...
11 and that 1 is any vector in JR
...
, b such that
11
with respect to an orthonormal basis
...
1)

If :B were an arbitrary basis, the procedure would be to solve the resulting system of
n equations in n variables
...
1) with vi to get

x · v;
because vi· v1

=

0 for i

=

o

+


...
+ o + biCvi ·vi)+ o + · · · + o

...
The result of this argument is important

enough to summarize as a theorem
...
, Vn}

is an orthonormal basis for JR
...
12 with respect to :Bis

b;
It follows that 1 can be written as

=

x ·v;

EXAMPLE4

2

Find the coordinates of 1

13

=

=

with respect to the orthonormal basis

3
4

{� 1�

1 1 -1
1'-
...


1 -1
2 1 '2
-1

Solution: By Theorem 2, the coordinates

bi
b2

=

=

1

·

v1

1
...
+2 + O
3

1

=

=

=

1

0

3

0

4)

-

4)

=

5

=

-1

3

-

0)

4)

=

=

-
...


Thus the 13-coordinate vector of 1 is [1]23

(It is easy to check that

- -
...


=

Another technical advantage of using orthonormal bases is related to the first one
...
If the
basis is not orthonormal, the calculations are a little ugly, but they are quite simple

{v1,
Y1V1 +

when the basis is orthonormal
...
n and let
+ YnVn be any vectors in JR"
...
• ,

·

· ·

that

vi· Vj

=

1· Y

0 for i * j and vi· v;
=

=

1 gives

(X1V1 + ''' + XnVn) (y1V1

+ ''' +

Y11V11)

X1Y1(V1 · V1) + X1Y2(V1 · V2) + · + X1Y11(V1 · Vn) + X2Y1(V2 V t )
+ · + X2Yn(V2 · V11) +
+ X11Y11(V11 Vn)
·

·

·

X1Y1
·

=

=

·

+

X2Y2

+
...
This fact will be used in Section 7 2

Thus, the formulas in the new coordinates look exactly like the formulas in standard


...
Ul, [-m
...
[i'
,
J
...

[ ] [
·

Verify the result of Example 5 by finding

1· y

i'

,Y

E

R3 such that

andt · jl
...
Ul

=

=

1

Ul Hl
·

=

=

[i']

o

=

Ul

and

6

-2

and y explicitly and computing

111112

and

directly
...


11

Dot products and orthonormal bases in JR
...

Section 7
...


inner

These will be considered in

Change of Coordinates and Orthogonal Matrices
A third technical advantage of using orthonormal bases is that it is very easy to invert

To keep the writing short, we give the argument in JR
...

works in any dimension
...


From Section 4
...
Now, consider the product of the transpose
of P with P: the rows of

pT are vf,vI, v;, so

viv2 viv3
viv2 viv3
vrv2 vrv3
Vt ·v2 Vt ·v3
v2 v2 v2 v3
v3 v2 v3 ·v3
·

·

·

1

Remark
We have used the fact that the matrix multiplication xT y equals the dot product x · y
...

It follows that if P is a square matrix whose columns are orthonormal, then P is
invertible and

Definition
Orthogonal Matrix

p-t pT
...

=

An n x n matrix P such that

p-t pT and that ppT
=

=

I

pTP
pTP
...
It follows that

=

It is important to observe that the definition of an orthogonal matrix is equivalent
to the orthonormality of either the columns or the rows of the matrix
...

(2) The columns of P form an orthonormal set
...


Proof: Let P

=

[vt

v,,J
...
But this is true if

and only if the columns of P form an orthonormal set
...


ppT

=

I
...
Be sure that you remember that
an orthogonal matrix has orthonormal columns and rows
...
e
sm

EXAMPLE 7
The set

{[��� ;] [-:�: :]}
]
,

is orthonormal for any e (verify)
...

cose

-[

T

...


{ [il [=: l H]}
·

·

_

]

cose

sine

- sm

...


is orthogonal
...
) If the vectors are normalized,

the resulting set is orthonormal, so the following matrix Pis orthogonal:

p

=

[

1/
...
-1/Y3 -1/\/6
2/\/6
0
-1/Y3

]

Thus, Pis invertible and

-l
p

=

T
p

=

[

1/
...

l/-fJ -1/Y3 -1 Y3
11\16 -11\16 2/\/6

�

Moreover, observe that p-I is also orthogonal
...

0
1/2 -1/2
1/
...

1/2
0
1/2 -1/2
-1/
...

1/
...


The most important application of orthogonal matrices considered in this book is
the diagonalization of symmetric matrices in Chapter 8, but there are many other geo­
metrical applications as well
...
(This was one question we could not answer in
Chapter 3
...

(2) 13 is orthonormal
...

Let us find such a basis
...
= 11:1 = },

[; l

We must find two vectors that are orthogonal to

/
...
Solving the equation

1
0 = f1 X = (X1 + X2 + X3)
Y3
-+

-+

·

by inspection, we find that the vector

[-i]

is orthogonal to

/
...
) To form a right­
handed system, we can now take the third vector to be

Normalizing these vectors, we get

and

The required right-handed orthonormal basis is thus 13

=

u7, fi, h},

and the

orthogonal change of coordinates matrix from this basis to the standard basis is

P

=[A

Ii

[
1/vf3 1/-Y'i
...

1/Y3

]

1/
...
/6
0 -2/
...
Notice
also that the matrix L
[ ]s is itself an orthogonal matrix
...


A Note on Rotation Transformations and Rotation of
Axes in R2

[
{ [���;] ,[-���:]}


...


Stne

]

- sine
...


...

Treatments of rotation of axes often emphasize the change of coordinates equation
...
4
...
Then, for any
x E JR2 with [x]1l

=

[:J

the change of coordinates equation can be written in the form

p-1 [x]s
...
Thus, the change of coordinates equation for this rotation of axes

[x]1l

=

can be written as

[:�] -[ ��� :
_

��� �][��]

Or it can be written as two equations for the "new" coordinates in terms of the "old":

b1
b2

=

=

X1 COS 8 + X2 sin 8

-X1 sin 8 + X2 COS 8

These equations could also be derived using a fairly simple trigonometric argument
...

cose - sine

...


...

a SO appeared tn ect10n 3
...
Conceptually, this is quite different from a rotation of axes
...
In fact, what may seem even more confusing is the fact that if y ou re­
place e with ( -e), one matrix turns into the other (because cos ( -e)
sin (-B)

=

=

cos e and

- sin B)
...
First, consider R to be the transformation
that rotates each vector by angle e, with 0 < e <

�;

then R(e1) is a vector in the

first quadrant that makes an angle e with the x1 -axis
...
Then e1 has not moved
but the axes have, and with respect to the new axes, e1 is in the new first quadrant and
makes an angle of e with respect to the Y1 -axis
...

Compare Figures 7
...
1 and 7
...
2
...
1
...
1
...
2

--+

IR
...


The standard basis vector e1 is shown relative to axes obtained from the
standard axis by rotation through -e
...
1
Practice Problems
Al Determine which of the following sets are orthog­

A4 For each of the following matrices, decide whether

responding orthonormal set and the orthogonal

orthogonal, indicate how the columns of A fail to

onal
...

(a)

(b)

(c)

(d)

{[ � ] [ � ] }

form an orthonormal set (for example, "the second
and third columns are not orthogonal")
...
[J HJ}
-21
{ 1� _-1: ' -�-01 ' 01}
{ Hl [�] {!]}

(a) A=
(b) A=
(c) A=
(d) A=

,

A2 �t S =

l

·

l

(e) A=
Find the coordi-

nates of each of the following vectors with respect

to the orthonormal basis :B
...
If A is not

:B

m
Hl
=

(b)t=

(d) Z=

1-1
2 11 -1
{' :
1

'2

1

' Y2

Hl
m

-10
10 -1!}
-14
3-5
-13
2-3
1

' Y2

Find the coordinates of each of the following vec-

tors with respect to the orthonormal basis :B
...
3

�

JR
...
§1
gi/l g;l 1, 2, 3,

nents of

(b) Let

-+

f;

:B =

_

g,

=

=

g3

-+

for

-+

{/i, h
...


t

=

g,

so that

is an orthonormal basis
...



...


For part (d),

it is probably easiest to write this in the form

[L]2l=

}i["
...


{[ -�;� l·[ �;� l [ � ]}
l/

A6 Given that '13=

1/
...
/6

·

-1/Y2

3
is an orthonormal basis for IR
...
that includes the vector

1/
...
/3
1;Y2

l

and briefly explain why your basis is

orthonormal
...
For each orthogonal set, produce the cor­
responding orthonormal set and the orthogonal

2

3

Bl Determine which of the following sets are ortho­
(a) w=

-2

(b) 1=

6

-4
0
4

change of coordinates matrix P
...
Ul [�J}
{[ll {�l HJ}

B2 Ut � =

0

1

}

form an orthonormal set (for example, "the second
and third columns are not orthogonal")
...

(a) w=

(c )

=

Y

B 3 Let '13 =

(b) x=

(d)Z=

{d

1


...
If A is not

2

},

(d ) z=

2
-2

B4 For each of the following matrices, decide whether

-1 2
0

4

5
0

: � !

-1

-

,�

,

0

},

nJ
nl}
·

Find

-1

the coordinates of each of the following vectors
with respect to the orthonormal basis '13
...
/3 l/Y2 -1/
...
/3 l/-f2
...
/6
2/
...
/3 0
1/
...
/6 l/Y2
1/
...
/6 l/Y2
1/
...
/6 0

BS (a) Ut W, =

[i]

[ �]
-

and W, =

·Determine a thfrd

vector w3 such that {w1, w2, w3} forms a right­
handed orthogonal set
...
ni
...
Find

and

Conceptual Problems
Dl Verify that the product of two orthogonal matrices

is an orthogonal matrix
...

(b) Give an example of a 2 x 2 matrix A such that
det A = 1, but A is not orthogonal
...


D3 (a) Use the fact that x

·

(b) Show that any real eigenvalue of an orthogonal
matrix must be either 1 or -1
...




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