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Chapter 5: Q24E (page 233)

If A and B are arbitraryn脳n matrices, which of the matrices in Exercise 21 through 26 must be symmetric?

A螕BA.

Short Answer

Expert verified

The matrixATBA is not symmetric.

Step by step solution

01

Condition to be a symmetric.

A matrix is symmetric if and only if it is equal to its transpose.

All entries above the main diagonal of a symmetric matrix are reflected into equal entries below the diagonal. A matrix is skew-symmetric if and only if it is the opposite of its transpose.

For example,

11-1120-105

Thus, it gives

ATBAT=BATATT=ATBTA

Therefore, if A and B are arbitrary n脳nmatrices then the matrices ATBAis not symmetric.

Hence, the matrix is not necessarily symmetric.

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Most popular questions from this chapter

TRUE OR FALSE?If matrix A is orthogonal, then the matrixAT must be orthogonal as well.

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? [-0.80.60.60.8].

Consider a linear transformationL from RmtoRn that preserves length. What can you say about the kernel of L? What is the dimension of the image? What can you say about the relationship between n and m? If Ais the matrix of L, What can you say about the columns of A? What isATA? What aboutAAT? Illustrate your answer with an example where m=2and n=3.

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

3.v鈬赌=[2345]

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