14. Q = /5 16. Q = 2= [₁ 1-2 21 32 -2 3

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ISBN:9780470458365
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Linear algebra: please solve q14 and 16 correctly and handwritten condition 5 are attached 

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]
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.
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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