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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: Q5E (page 233)

If the $n 脳 n$ matrices Aand Bare orthogonal, which of the matrices in Exercise 5 through 11 must be orthogonal as well?3A.

Short Answer

Expert verified

The Matrix 3A is not orthogonal.

Step by step solution

01

Definition of Orthogonal transformation

An $n 脳 n$ matrix Ais orthogonal if and only if its columns form an orthonormal basis oflocalid="1659490737513" ${鈩�}^{n}$ .

02

Verification whether the given matrix is orthogonal.

Given that A and B are orthogonal matrices.

Let $A = [{\overset{鈫�}{v}}_{1} {\overset{鈫�}{v}}_{2} . . . {\overset{鈫�}{v}}_{n}]$ where $i 鈮� j$ .

Then, ${\overset{鈫�}{V}}_{i} - \overset{鈫�}{V_{j}} = 0$ and $||\overset{鈫�}{v_{i}}|| = 1$ for $i = 1, 2, . . ., n$ .

Therefore, it gives $3 A = [3 {\overset{鈫�}{v}}_{1} 3 {\overset{鈫�}{v}}_{1} . . . 3 {\overset{鈫�}{v}}_{n}]$ .

The columns are orthogonal but the length is not 1.

$||3 {\overset{鈫�}{v}}_{i}|| = 3 ||{\overset{鈫�}{v}}_{i}|| = 3$

Thus,3A is not an orthogonal matrix.

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Most popular questions from this chapter

Question: If the $n 脳 n$ matrices Aand Bare symmetric and Bis invertible, which of the matrices in Exercise 13 through 20 must be symmetric as well?role="math" localid="1659492178067" $B^{- 1}$ .

Are the rows of an orthogonal matrix A necessarily orthonormal?

The formula $A {(A^{T} A)}^{- 1}$ for the matrix of an orthogonalprojection is derived in Exercise 67. Now considerthe QRfactorization of A, and express the matrix $A {(A^{T} A)}^{- 1} A^{T}$ in terms of Q.

Using paper and pencil, perform the Gram-Schmidt process on the sequences of vectors given in Exercises 1 through 14.

$4 路 [\begin{matrix} 4 \\ 0 \\ 3 \end{matrix}], [\begin{matrix} 25 \\ 0 \\ - 25 \end{matrix}], [\begin{matrix} 0 \\ - 2 \\ 0 \end{matrix}]$

For each pair of vectors $\overset{鈫�}{u}$ and $\overset{鈫�}{v}$ listed in Exercises 7 through 9, determine whether the angle $胃$ between $\overset{鈫�}{u}$ and $\overset{鈫�}{v}$ is acute, obtuse, or right.

9. $\overset{鈫�}{u} = [\begin{matrix} 1 \\ - 1 \\ 1 \\ - 1 \end{matrix}], \overset{鈫�}{v} = [\begin{matrix} 3 \\ 4 \\ 5 \\ 3 \end{matrix}]$ .

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