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Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 5: Q22E (page 233)

If A and B are arbitrary $n 脳 n$ matrices, which of the matrices in Exercise 21 through 26 must be symmetric?
$B B^{T}$ .

Short Answer

Expert verified

The matrix $B B^{T}$ 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,

Let the matrix is $B = (\begin{matrix} 1 & 1 & - 1 \\ 1 & 2 & 0 \\ - 1 & 0 & 5 \end{matrix})$ .

Thus, it gives

${(B B^{T})}^{T} = {(B^{T})}^{T} B^{T} = B B^{T}$

Thus, If A and B are arbitrary $n 脳 n$ matrices then the matrices $B B^{T}$ is symmetric.

Hence, the matrices is symmetric.

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Most popular questions from this chapter

Using paper and pencil, perform the Gram-Schmidt process on the sequences of vectors given in Exercises 1 through 14.

5. $[\begin{matrix} 2 \\ 2 \\ 1 \end{matrix}], [\begin{matrix} 1 \\ 1 \\ 5 \end{matrix}]$

Consider a basis ${\overset{鈫�}{v}}_{1}, {\overset{鈫�}{v}}_{2}, . . . {\overset{鈫�}{v}}_{m}$ of a subspaceVofrole="math" localid="1659434380505" ${鈩�}^{n}$ . Show that a vector $\overset{鈫�}{x}$ inrole="math" localid="1659434402220" ${鈩�}^{n}$ is orthogonal toV if and only if $\overset{鈫�}{x}$ is orthogonal to all vectors ${\overset{鈫�}{v}}_{1}, {\overset{鈫�}{v}}_{2}, . . . {\overset{鈫�}{v}}_{m}$ .

For each pair of vectors $\overset{鈫�}{u}$ and $\overset{鈫�}{v}$ listed in Exercises 7 through 9, determine whether the angle $胃$ between $\overset{鈫�}{u}$ and $\overset{鈫�}{v}$ is acute, obtuse, or right.

8. $\overset{鈫�}{u} = [\begin{matrix} 2 \\ 3 \\ 4 \end{matrix}], \overset{鈫�}{v} = [\begin{matrix} 2 \\ - 8 \\ 5 \end{matrix}]$

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? $\frac{1}{7} [\begin{matrix} 2 & 6 & - 3 \\ 6 & - 3 & 2 \\ 3 & 2 & 6 \end{matrix}]$ .

a.Consider a vector $\overset{鈯�}{v}$ in $R^{n}$ , and a scalar k. Show that $| | k \overset{鈫�}{v} | | = | k | . | | \overset{鈫�}{v} | |$

b.Show that if $\overset{鈯�}{v}$ is a nonzero vector in $R^{n}$ , then

$\overset{鈫�}{u} = \frac{1}{| | \overset{鈫�}{v} | |}$ is a unit vector.

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