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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 8: Q13E (page 413)

If the singular values of a $2 脳 2$ matrix $A$ are 3 and 4, then there must exist a unit vector $\overset{鈫�}{u}$ in $R^{2}$ such that $|| A \overset{鈫�}{u} || = 4$ .

Short Answer

Expert verified

The given statement is TRUE.

Step by step solution

01

Definition of a singular value decomposition

The single value decomposition produces orthonormal bases of v鈥檚 and u鈥檚 for the four fundamental sub-spaces.

Using those bases, A becomes a diagonal matrix $鈭�$ and $A v_{i} = 蟽_{i} u_{i}$ , $蟽_{i}$ = singular value.

02

Step 2: To Find TRUE or FALSE

By the standard notation of the singular value decomposition, $蟽_{1} = 4$ and $A {\overset{藱}{i}}_{1} = 蟽_{1} {\overset{藱}{u}}_{1} = 4 {\overset{藱}{u}}_{1}$ .

Both ${\overset{藱}{v}}_{1}$ and ${\overset{藱}{u}}_{1}$ are the unit vectors. Thus,

$||A {\overset{藱}{v}}_{1}|| = ||蟽_{1} {\overset{藱}{u}}_{1}|| ||A {\overset{藱}{v}}_{1}|| = 4 ||{\overset{藱}{u}}_{1}|| ||A {\overset{藱}{v}}_{1}|| = 4$

Thus, the given statement isTRUE.

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Most popular questions from this chapter

51. IfAis a symmetric $2 脳 2$ matrix with eigenvalues 1 and 2, then the angle between $\overset{鈫�}{x}$ and $A \overset{鈫�}{x}$ must be less than $蟺 / 6$ , for all nonzero vectors $\overset{鈫�}{x}$ in $R^{2}$ .

For ndistinct scalars $a_{1}, a_{2}, . . ., a_{n}$ , find

role="math" localid="1659609991385" $d e t [\begin{matrix} a_{1} & a_{2} & 鈥� & a_{n} \\ a_{1}^{2} & a_{2}^{2} & 鈥� & a_{n}^{2} \\ 鈰� & 鈰� & 鈰� \\ a_{1}^{n} & a_{2}^{n} & 鈥� & a_{n}^{n} \end{matrix}]$

Let Abe anmatrix and $\overset{鈫�}{v}$ a vector in $R^{m} .$ Show that

$蟽_{m} || \overset{鈫�}{v} || 鈮� || A \overset{鈫�}{v} || 鈮� 蟽_{1} || \overset{鈫�}{v} ||$

where $蟽_{1} a n d 蟽_{m}$ are the largest and the smallest singular values of A, respectively. Compare this with Exercise 25.

For the matrix $A = [\begin{matrix} 8 & - 2 \\ - 2 & 5 \end{matrix}],$ write $A = B B^{T}$ as discussed in Exercise 28. See Example 1.

If Ais any symmetric 2x2matrix with eigenvalues -2 and 3, and $\overset{鈫�}{u}$ is a unit vector ${鈩�}^{2}, {鈩�}^{2}$ , what are the possible values of the dot product $\overset{鈫�}{u} 脳 A \overset{鈫�}{u}$ ? Illustrate your answer, in terms of the unit circle and its image A.

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