Refer a friend and earn $50 when they subscribe to an annual planRefer Now

Get the answer to your homework problem.

Try Numerade Free for 30 Days

Like

Report

$A$ is an $m \times n$ matrix with a singular value decomposition $A=U \Sigma V^{T},$ where $U$ is an $m \times m$ orthogonal matrix, $\Sigma$ is an $m \times n$ 'diagonal" matrix with $r$ positive entries and no negative entries, and $V$ is an $n \times n$ orthogonal matrix. Justify each answer.

Justify the statement in Example 2 that the second singular value of a matrix $A$ is the maximum of $\|A \mathbf{x}\|$ as $\mathbf{x}$ varies over all unit vectors orthogonal to $\mathbf{v}_{1},$ with $\mathbf{v}_{1}$ a right singular vector corresponding to the first singular value of $A .[\text { Hint: }$ Use Theorem 7 in Section $7.3 . ]$

The square root of $\lambda_{2},$ which is the second largest singular value of $A,$ is the maximum of$\|A X\|$ overall unit vectors orthogonal to $v_{1}$ -

Algebra

Chapter 7

Symmetric Matrices and Quadratic Forms

Section 4

The Singular Value Decomposition

Introduction to Matrices

Missouri State University

Oregon State University

Lectures

01:32

In mathematics, the absolu…

01:11

02:09

$A$ is an $m \times n$ mat…

01:25

05:05

06:04

02:32

02:23

01:38

An $n \times n$ matrix $A$…

02:36

A real $n \times n$ matrix…

06:49

Okay, so this properly conceived there the men tricks a transpose times a, uh and they're supposed that a transposed time. Say, has, um I can barrios Lunda one greet the number two and the wouldn't recover by the rest. We only are discussed on the 1st 2 again values. And so Lambda Wise, The greatest again value and the love industrious. A second greatest rag ag value. Are we thanking vectors of you Wanted to be chew. Okay, so bye bye purpose he earned. If you're in section um his section 7.3, we know that if we can see that the quadratic form Q X equals toe extras post times a transpose a time sex If we can see that this quadratic form q x the truth, its maximum value if and only if x equals to be one. And we also know que eggs triffids maximum value, um, subject to eggs perpendicular to V one. He finally if x equals Toby too, which is the second Eigen vector corresponding to the second largest tagging value. Lambda too. So this proof, the statements in this case is cryogenic form queues off V two. He closed to me to transpose times, age, chance, post times a times V two equals to the norm off eight tons V two, which is Lambda two times of the two, The Simpsons. Well, seem that both B one B two are normalized victors. Um, so then use their norms are one. So this equals to dance out, too, which is the second largest I can value.

View More Answers From This Book

Find Another Textbook

Numerade Educator

In mathematics, the absolute value or modulus |x| of a real number x is its …

$A$ is an $m \times n$ matrix with a singular value decomposition $A=U \Sigm…

An $n \times n$ matrix $A$ that satisfies $A^{T}=A^{-1}$ is called an orthog…

A real $n \times n$ matrix $A$ is called orthogonal if $A A^{T}=$ $A^{T} A=I…

An $n \times n$ matrix $A$ is called orthogonal if $A^{T}=A^{-1}$ Show that …

08:05

Let $\mathbf{v}_{1}=\left[\begin{array}{l}{1} \\ {0}\end{array}\right], \mat…

04:47

Exercises $3-8$ refer to $\mathbb{P}_{2}$ with the inner product given by ev…

00:52

The space is $C[0,2 \pi]$ with the inner product $(6)$.Find the third-or…

07:43

Let $A$ be the matrix of the quadratic form$$9 x_{1}^{2}+7 x_{2}^{2}+11 …

01:20

Determine which of the matrices in Exercises $1-6$ are symmetric. $$…

01:36

Determine which of the matrices in Exercises $7-12$ are orthogonal. If ortho…

00:48

Determine which pairs of vectors in Exercises $15-18$ are orthogonal.$$<…

00:38

[M] Compute an SVD of each matrix. Report the final matrix entries accurate …

08:58

Let $\mathbf{u}$ be a unit vector in $\mathbb{R}^{n},$ and let $B=\mathbf{u}…

03:29

Consider the points in Exercise 6 in Section $8.1 .$ Which of $\mathbf{p}_{1…

92% of Numerade students report better grades.

Try Numerade Free for 30 Days. You can cancel at any time.

Annual

0.00/mo 0.00/mo

Billed annually at 0.00/yr after free trial

Monthly

0.00/mo

Billed monthly at 0.00/mo after free trial

Earn better grades with our study tools:

Textbooks

Video lessons matched directly to the problems in your textbooks.

Ask a Question

Can't find a question? Ask our 30,000+ educators for help.

Courses

Watch full-length courses, covering key principles and concepts.

AI Tutor

Receive weekly guidance from the world’s first A.I. Tutor, Ace.

30 day free trial, then pay 0.00/month

30 day free trial, then pay 0.00/year

You can cancel anytime

OR PAY WITH

Your subscription has started!

The number 2 is also the smallest & first prime number (since every other even number is divisible by two).

If you write pi (to the first two decimal places of 3.14) backwards, in big, block letters it actually reads "PIE".

Receive weekly guidance from the world's first A.I. Tutor, Ace.

Mount Everest weighs an estimated 357 trillion pounds

Snapshot a problem with the Numerade app, and we'll give you the video solution.

A cheetah can run up to 76 miles per hour, and can go from 0 to 60 miles per hour in less than three seconds.

Back in a jiffy? You'd better be fast! A "jiffy" is an actual length of time, equal to about 1/100th of a second.