Q4E Use the various characterization... [FREE SOLUTION]

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Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 5: Q4E (page 233)

Use the various characterizations of orthogonal transformations and orthogonal matrices. Find the matrix of an orthogonal projection. Use the properties of the transpose. Which of the matrices in Exercise 1 through 4 are orthogonal? $\frac{1}{7} [\begin{matrix} 2 & 6 & - 3 \\ 6 & - 3 & 2 \\ 3 & 2 & 6 \end{matrix}]$ .

Short Answer

Expert verified

The Matrix $\frac{1}{7} [\begin{matrix} 2 & 6 & - 3 \\ 6 & - 3 & 2 \\ 3 & 2 & 6 \end{matrix}]$ is not an orthogonal.

Step by step solution

Definition of Orthogonal.

A square matrix is orthogonal matrix if $A 脳 A^{T} = I .$

Verification whether the given matrix is orthogonal.

Let the given matrix is $A = \frac{1}{7} [\begin{matrix} 2 & 6 & - 3 \\ 6 & - 3 & 2 \\ 3 & 2 & 6 \end{matrix}]$ which is written as $A = [\begin{matrix} \frac{2}{7} & \frac{6}{7} & \frac{3}{7} \\ \frac{6}{7} & \frac{- 3}{7} & \frac{2}{7} \\ \frac{- 3}{7} & \frac{2}{7} & \frac{6}{7} \end{matrix}]$ .

Then, transpose of the given matrix is $A^{T} = [\begin{matrix} \frac{2}{7} & \frac{6}{7} & \frac{3}{7} \\ \frac{6}{7} & \frac{- 3}{7} & \frac{2}{7} \\ \frac{- 3}{7} & \frac{2}{7} & \frac{6}{7} \end{matrix}]$ .

Thus, for orthogonal matrix,

$\begin{aligned} A 脳 A^{鈯�} = [\begin{array}{ccc} \frac{2}{7} & \frac{6}{7} & \frac{鈭� 3}{7} \\ \frac{6}{7} & \frac{鈭� 3}{7} & \frac{2}{7} \\ \frac{3}{7} & \frac{2}{7} & \frac{6}{7} \end{array}] 脳 [\begin{array}{ccc} \frac{2}{7} & \frac{6}{7} & \frac{3}{7} \\ \frac{6}{7} & \frac{鈭� 3}{7} & \frac{2}{7} \\ \frac{鈭� 3}{7} & \frac{2}{7} & \frac{6}{7} \end{array}] \\ = [\begin{array}{cccc} \frac{4}{36} + \frac{36}{49} + \frac{9}{49} & \frac{12}{49} 鈭� \frac{18}{49} 鈭� \frac{6}{49} & \frac{6}{49} + \frac{12}{49} 鈭� \frac{18}{49} \\ \frac{12}{49} - \frac{18}{49} 鈭� \frac{6}{49} & \frac{36}{49} + \frac{9}{49} + \frac{4}{49} & \frac{18}{49} 鈭� \frac{6}{49} + \frac{12}{49} \\ \frac{6}{49} + \frac{12}{49} 鈭� \frac{18}{49} & \frac{18}{49} 鈭� \frac{6}{49} + \frac{12}{49} & \frac{9}{49} + \frac{4}{49} + \frac{36}{49} \end{array}] \end{aligned}$

$= [\begin{array}{ccc} 1 & \frac{鈭� 12}{49} & 0 \\ \frac{鈭� 12}{49} & 1 & \frac{24}{49} \\ 0 & \frac{24}{49} & 1 \end{array}]$

Since, the matrix not satisfy orthogonal condition it gives $A 脳 A^{T} 鈮� I .$ .

Hence,localid="1659509364173" $A = [\begin{matrix} \frac{2}{7} & \frac{6}{7} & \frac{- 3}{7} \\ \frac{6}{7} & \frac{- 3}{7} & \frac{2}{7} \\ \frac{3}{7} & \frac{2}{7} & \frac{6}{7} \end{matrix}]$ is not an orthogonal matrix.

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91影视

Short Answer

Step by step solution

Definition of Orthogonal.

Verification whether the given matrix is orthogonal.

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91影视

Use the various characterizations of orthogonal transformations and orthogonal matrices. Find the matrix of an orthogonal projection. Use the properties of the transpose. Which of the matrices in Exercise 1 through 4 are orthogonal? 17[26-36-32326].

Short Answer

Step by step solution

Definition of Orthogonal.

Verification whether the given matrix is orthogonal.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Decision Maths

Pure Maths

Statistics

Geometry

Discrete Mathematics

Study anywhere. Anytime. Across all devices.