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Chapter 5: Q11E (page 233)

If the nxnmatrices Aand Bare orthogonal, which of the matrices in Exercise 5 through 11 must be orthogonal as well? AT.

Short Answer

Expert verified

The MatrixAT is orthogonal.

Step by step solution

01

Definition of Orthogonal.

A square matrix is orthogonal matrix ifAXAT=I

02

Verification whether the given matrix is orthogonal.

Given that A and B are orthogonal matrices.

By theorem: In an nxn matrix, a matrix A is orthogonal ifA脳AT=In or, equivalently, if A1=AT.

By theorem: The inverse A1of an nxn matrix A is orthogonal.

By the above theorems, it can conclude thatAT matrix is orthogonal.

Hence,AT matrix is orthogonal.

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Most popular questions from this chapter

In Exercises 40 through 46, consider vectors v鈨�1,v鈨�2,v鈨�3in R4;; we are told that is the entry of matrix A.

localid="1659439944660" role="math" A=[35115920112049]

45. Find projv(v鈨�1),,where V=span (v鈨�2.v鈨�3).Express your answer as a linear combination of v鈨�2and v鈨�3.

In Exercises 40 through 46, consider vectors in v鈫�1,v鈫�2,v鈫�3; we are told thatrole="math" localid="1659495854834" v鈫�i,v鈫�j is the entryaij of matrix A.

A=[35115920112049]

46. Find projv(v鈫�3), where V =span role="math" localid="1659495997207" (v鈫�1.v鈫�2). Express your answer as a linear combination ofrole="math" localid="1659496026018" v鈫�1 and v鈫�2.

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

8.[5422],[367-2]

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

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].
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