Q2E Use the various characterization... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 5: Q2E (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? $[\begin{matrix} - 0.8 & 0.6 \\ 0.6 & 0.8 \end{matrix}]$ .

Short Answer

Expert verified

The Matrix $[\begin{matrix} - 0.8 & 0.6 \\ 0.6 & 0.8 \end{matrix}]$ is 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 $A = [\begin{matrix} - 0.8 & 0.6 \\ 0.6 & 0.8 \end{matrix}]$ .

Then, the transpose of the given matrix is $A^{T} = [\begin{matrix} 0.8 & 0.6 \\ 0.6 & 0.8 \end{matrix}]$ .

Thus, for orthogonal matrix,

$A 脳 A^{T} = [\begin{matrix} - 0.8 & 0.6 \\ 0.6 & 0.8 \end{matrix}] 脳 [\begin{matrix} - 0.8 & 0.6 \\ 0.6 & 0.8 \end{matrix}] = [\begin{matrix} - 0.8 & 脳 - 0.8 + \\ 0.6 & 脳 - 0.8 + \end{matrix} \begin{matrix} 0.6 & 0.6 \\ 0.8 & 0.6 \end{matrix} \begin{matrix} - 0.8 & 0.6 + \\ 0.6 & 0.6 + \end{matrix} \begin{matrix} 0.6 & 0.8 \\ 0.8 & 0.8 \end{matrix}] = [\begin{matrix} 0.64 + & 0.36 \\ - 0.48 + & 0.48 \end{matrix} \begin{matrix} - 0.48 + & 0.48 \\ 0.36 + & 0.64 \end{matrix}] = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$

Since, the matrix satisfies orthogonal condition $A 脳 A^{T} = I$

Hence, $A = [\begin{matrix} 0.6 & 0.8 \\ 0.8 & 0.6 \end{matrix}]$ is an orthogonal matrix.

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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? $[\begin{matrix} - 0.8 & 0.6 \\ 0.6 & 0.8 \end{matrix}]$ .

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? [-0.80.60.60.8].

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

Applied Mathematics

Pure Maths

Discrete Mathematics

Logic and Functions

Statistics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

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? $[\begin{matrix} - 0.8 & 0.6 \\ 0.6 & 0.8 \end{matrix}]$ .