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Chapter 5: Q8E (page 233)

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

Short Answer

Expert verified

The Matrix A+B is not orthogonal.

Step by step solution

01

Definition of Orthogonal transformation

Ann脳nmatrix Ais orthogonal if and only if its columns form an orthonormal basis of鈩�n

02

Verification whether the given matrix is orthogonal.

Given that AandB are orthogonal matrices.

That is for anyv鈫�鈭�鈩�nit has鈥�Av鈫�鈥�=鈥�v鈫�鈥� and 鈥�Bv鈫�鈥�=鈥�v鈫�鈥�.

鈥�A+Bv鈫�鈥�=鈥�-Av鈫�+Bv鈫�鈥�=鈥�Av鈫�鈥�Bv鈫�=v鈫�+v鈫�=2v鈫�

As the length of each column vector is 2, A+B is not an orthogonal matrix.

Hence, the Matrix A+B is not orthogonal.

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Most popular questions from this chapter

Prove Theorem 5.1.8d.(V鈯�)鈯�=Vfor any subspaceV of鈩�n.

Consider the vectors

ur=[11...1]andvr=[11...1]inRn

a.For n =2,3,4 , find the angle胃between role="math" localid="1659432008110" u鈯�and v鈯�. For n=2and 3, represent the vectors graphically.

b.Find the limit of 胃as napproaches infinity.

Let Abe the matrix of an orthogonal projection. FindA2 in two ways:

a.Geometrically. (Consider what happens when you apply an orthogonal projection twice.)

b.By computation, using the formula given in Theorem 5.3.10

Are the rows of an orthogonal matrix A necessarily orthonormal?

Find the length of each of the vectorsv鈬赌In exercises 1 through 3.

3.v鈬赌=[2345]

See all solutions

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