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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: Q10E (page 233)

If the nxnmatrices Aand Bare orthogonal, which of the matrices in Exercise 5 through 11 must be orthogonal as well? $B^{- 1} AB$ .

Short Answer

Expert verified

The Matrix $B^{- 1} AB$ is orthogonal.

Step by step solution

01

Definition of Orthogonal.

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

02

Verification whether the given matrix is orthogonal.

Given that A and B are orthogonal matrices.

Then by properties of transpose, ${(A B)}^{T} = B^{T} A^{T}$ it has

$(B^{1}) | (B^{1} A)) = (B (A ((B^{1})) (B^{1}]$

By theorem: In an matrix, a matrix A is orthogonal if or, equivalently, if $A 脳 A^{T} = I_{n}$ . $A^{1} = A^{T}$

$(B^{1} A B) ((B' A)) = (B (A ((B^{I})) (B^{I})$

Since, it is written as $A^{1} = A^{T}$ .

$(B^{1} A B) ((B' A)) = (B (A ((B^{I})) (B^{I})$

By association property,

$(B^{|} A B) ((B (R)) = B ((A ((B^{I}))$

Then, it is observed as,

${(B^{- 1} A B)}^{T} (B^{- 1} A B) = B^{T} (A^{T} (I_{n}) A) B = B^{T} (A^{T} A) B = B^{T} (I_{n}) B = B^{T} B = I_{n}$

Therefore, $B^{- 1} A B$ matrix is orthogonal.

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Most popular questions from this chapter

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

Complete the proof of Theorem 5.1.4: Orthogonal projection is linear transformation.

Find the length of each of the vectors $\overset{鈬赌}{v}$ In exercises 1 through 3.

3. $\overset{鈬赌}{v} = [\begin{matrix} 2 \\ 3 \\ 4 \\ 5 \end{matrix}]$

Prove Theorem 5.1.8d. ${(V^{鈯�})}^{鈯�} = V$ for any subspaceV of ${鈩�}^{n}$ .

Using paper and pencil, find the QR factorization of the matrices in Exercises 15 through 28. Compare with Exercises 1 through 14.

16. $[\begin{matrix} 6 & 2 \\ 3 & - 6 \\ 2 & 3 \end{matrix}]$ .

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