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Chapter 5: Q26E (page 233)

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

B(A+A螕)B螕.

Short Answer

Expert verified

B(A+A螕)B螕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.

For example,

11-1120-105

Let C=BA+ATBTnow the transpose of C is found as.

C=BA+ATTBT鈬�CT=BAT+AT鈬�C=CT

Hence, the matrix is symmetric.

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Most popular questions from this chapter

(a) Consider an matrix A such that A螕A=Im. It is necessarily true that? Explain.

(b) Consider an n脳nmatrix A such that ATA=In. Is it necessarily true that AAT=In? Explain.

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.

For which value(s) of the constant k are the vectors

u鈫�=[234]andv鈫�[1k1]and perpendicular?

Among all the vectors in Rnwhose components add up to 1, find the vector of minimal length. In the casen=2 , explain your solution geometrically.

Find the angle胃between each of the pairs of vectors u鈫�and localid="1659433601917" v鈫�in exercises 4 through

6.u鈫�=[1-12-2,v鈫�=2345]
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