14. Q = /5 16. Q = 2= [₁ 1-2 21 32 -2 3
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![In Exercises 13-18, use condition (5) to determine
whether the given matrix Q is orthogonal.
01
0
-[!]
[22]
13. Q =
15. Q=
14. Q =
16. Q
[√√3 1 √2]
=
1
[27]
1
1
√5
[23]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7bf79e7e-db7e-4b5c-a300-38b09014f3cb%2F68207888-5ece-45c8-b4b8-4f399b1514c3%2Frj2qb3_processed.jpeg&w=3840&q=75)
Transcribed Image Text:In Exercises 13-18, use condition (5) to determine
whether the given matrix Q is orthogonal.
01
0
-[!]
[22]
13. Q =
15. Q=
14. Q =
16. Q
[√√3 1 √2]
=
1
[27]
1
1
√5
[23]
![DEFINITION 9
Orthogonal Matrices
A remarkable and useful fact about symmetric matrices is that they are always diago-
nalizable. Moreover, the diagonalization of a symmetric matrix A can be accomplished
with a special type of matrix known as an orthogonal matrix.
EXAMPLE 5
Solution
A real (n x n) matrix Q is called an orthogonal matrix if Q is invertible and
Q-¹ = QT.
4.7 Similarity Transformations and Diagonalization
Definition 9 can be rephrased as follows: A real square matrix Q is orthogonal if
and only if
Q¹Q = 1.
(4)
Another useful description of orthogonal matrices can be obtained from Eq. (4). In
particular, suppose that Q = [91, 92, ..., qn] is an (n × n) matrix. Since the ith row of
Q¹ is equal to q, the definition of matrix multiplication tells us:
The ijth entry of QTQ is equal to qq.
Therefore, by Eq. (4), an (n × n) matrix Q = [9₁, 92, ..., qn] is orthogonal if and only
if:
Q₁ =
Verify that the matrices, Q₁ and Q2 are orthogonal:
1
0 1
0 √20
-1 0 1
1
√√2
The columns of Q, {91, 92,. .., qn},
form an orthonormal set of vectors.
0
1
and Q₂ =
0
1
0 0
10
1
0
=
We use Eq. (4) to show that Q₁ is orthogonal. Specifically,
1 0 -1
1 0 1
of Q₁
Q₁ =
√2
0 √20
0
-1 0 1
We use condition Eq. (5) to show that Q₂ is orthogonal. The column vectors of Q₂ are,
in the order they appear, {e2, e3, e₁}. Since these vectors are orthonormal, it follows
from Eq. (5) that Q2 is orthogonal.
1
0
0
1
2
331
20 0
020
0 0 2
= I.
(5)
From the characterization of orthogonal matrices given in condition Eq. (5), the
following observation can be made: If Q = [q1, 92, … , ¶] is an (n × n) orthogonal
matrix and if P = [P₁, P2, ..., Pn] is formed by rearranging the columns of Q, then P
is also an orthogonal matrix.
As a special case of this observation, suppose that P is a matrix formed by rear-
ranging the columns of the identity matrix, I. Then, since I is an orthogonal matrix,
it follows that P is orthogonal as well. Such a matrix P, formed by rearranging the
columns of I, is called a permutation matrix. The matrix Q₂ in Example 5 is a specific
instance of a (3 x 3) permutation matrix.
Orthogonal matrices have some special properties that make them valuable tools
for applications. These properties were mentioned in Section 3.7 with regard to (2 x 2)
orthogonal matrices. Suppose we think of an (n xn) matrix Q as defining a function (or
linear transformation) from R" to R". That is, for x in R", consider the function defined
by
y = Qx.
As the next theorem shows, if Q is orthogonal, then the function y = Qx preserves the
lengths of vectors and the angles between pairs of vectors.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7bf79e7e-db7e-4b5c-a300-38b09014f3cb%2F68207888-5ece-45c8-b4b8-4f399b1514c3%2Fs0wrv8t_processed.jpeg&w=3840&q=75)
Transcribed Image Text:DEFINITION 9
Orthogonal Matrices
A remarkable and useful fact about symmetric matrices is that they are always diago-
nalizable. Moreover, the diagonalization of a symmetric matrix A can be accomplished
with a special type of matrix known as an orthogonal matrix.
EXAMPLE 5
Solution
A real (n x n) matrix Q is called an orthogonal matrix if Q is invertible and
Q-¹ = QT.
4.7 Similarity Transformations and Diagonalization
Definition 9 can be rephrased as follows: A real square matrix Q is orthogonal if
and only if
Q¹Q = 1.
(4)
Another useful description of orthogonal matrices can be obtained from Eq. (4). In
particular, suppose that Q = [91, 92, ..., qn] is an (n × n) matrix. Since the ith row of
Q¹ is equal to q, the definition of matrix multiplication tells us:
The ijth entry of QTQ is equal to qq.
Therefore, by Eq. (4), an (n × n) matrix Q = [9₁, 92, ..., qn] is orthogonal if and only
if:
Q₁ =
Verify that the matrices, Q₁ and Q2 are orthogonal:
1
0 1
0 √20
-1 0 1
1
√√2
The columns of Q, {91, 92,. .., qn},
form an orthonormal set of vectors.
0
1
and Q₂ =
0
1
0 0
10
1
0
=
We use Eq. (4) to show that Q₁ is orthogonal. Specifically,
1 0 -1
1 0 1
of Q₁
Q₁ =
√2
0 √20
0
-1 0 1
We use condition Eq. (5) to show that Q₂ is orthogonal. The column vectors of Q₂ are,
in the order they appear, {e2, e3, e₁}. Since these vectors are orthonormal, it follows
from Eq. (5) that Q2 is orthogonal.
1
0
0
1
2
331
20 0
020
0 0 2
= I.
(5)
From the characterization of orthogonal matrices given in condition Eq. (5), the
following observation can be made: If Q = [q1, 92, … , ¶] is an (n × n) orthogonal
matrix and if P = [P₁, P2, ..., Pn] is formed by rearranging the columns of Q, then P
is also an orthogonal matrix.
As a special case of this observation, suppose that P is a matrix formed by rear-
ranging the columns of the identity matrix, I. Then, since I is an orthogonal matrix,
it follows that P is orthogonal as well. Such a matrix P, formed by rearranging the
columns of I, is called a permutation matrix. The matrix Q₂ in Example 5 is a specific
instance of a (3 x 3) permutation matrix.
Orthogonal matrices have some special properties that make them valuable tools
for applications. These properties were mentioned in Section 3.7 with regard to (2 x 2)
orthogonal matrices. Suppose we think of an (n xn) matrix Q as defining a function (or
linear transformation) from R" to R". That is, for x in R", consider the function defined
by
y = Qx.
As the next theorem shows, if Q is orthogonal, then the function y = Qx preserves the
lengths of vectors and the angles between pairs of vectors.
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