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Chapter 5: Q21E (page 233)

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

ATA.

Short Answer

Expert verified

ATA is 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.

02

Example

For example,

Let A=11-1120-105.

Thus, it gives

ATAT=ATATT=ATA

Thus, If A and B are arbitrary n脳n matrices then the matricesATA is symmetric.

Hence, the matrices is symmetric.

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Most popular questions from this chapter

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?AB2A.

Let Abe ann脳mmatrix. Is the formula(kerA)鈯�=im(AT)necessarily true? Explain.

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

20.[235046007]

a.Give an example of a (nonzero) skew-symmetric 3脳3 matrix A, and compute.

b.If an n脳nmatrix Ais skew-symmetric, is matrix A2necessarily skew-symmetric as well? Or is A2necessarily symmetric?

a.Find all n脳nmatrices that are both orthogonal and upper triangular, with positive diagonal entries.

b.Show that the QRfactorization of an invertible n脳nmatrix is unique. Hint: If,A=Q1R1=Q2R2 thenthe matrixA=Q2-1Q1=Q2R1-1 is both orthogonal and upper triangular, with positive diagonal entries.
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