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Chapter 5: Q45E (page 264)

TRUE OR FALSE

If A is an n脳nmatrix such that||Au鈫�||=1 for all unit vectors u鈫�, then A must be an orthogonal matrix.

Short Answer

Expert verified

The given statement is true.

Step by step solution

01

Definition of orthogonal matrix

Matrix A is orthogonal if and only ifx鈭�Rn,Ax鈫�=x鈫�.

02

Explanation for the given statement

Consider the statement as follows:

鈥淚f A is ann脳nmatrix such that Au鈫�=1for all unit vector u鈫�, then A must be an orthogonal matrix.

Let u鈫�is a unit vector such that Au鈫�=1.

Then for any x鈭�Rn, u鈫�=x鈫�x鈫�.

Au鈫�=1Ax鈫�x鈫�=11x鈫�Ax鈫�=1Ax鈫�=x鈫�

Hence, A is orthogonal matrix.

Thus, the given statement 鈥淚f A is an n脳nmatrix such thatAu鈫�=1 for all unit vector u鈫�, then A must be an orthogonal matrix鈥� is true.

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Most popular questions from this chapter

If the nxn matrices Aand Bare symmetric and Bis invertible, which of the matrices in Exercise 13 through 20 must be symmetric as well? -B.

Question: If the n脳nmatrices Aand Bare symmetric and Bis invertible, which of the matrices in Exercise 13 through 20 must be symmetric as well?A+B.

If A is an n脳nmatrix such that role="math" localid="1659514225617" AAT=In, then A must be an orthogonal matrix.

TRUE OR FALSE?If x鈬赌andy鈬赌are two vectors inRn, then the equation role="math" localid="1659506190737" 鈭�x鈫�+y鈫�鈭�2=鈭�x鈫�鈭�2+鈭�y鈫�鈭�2must hold.

Question:TRUE OR FALSE?If matrices A and Bare commute, then A must commuteBT withas well.
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