a-is-an-m-times-n-matrix-with-a-singular-value-decomposition-au-sigma-vt-where-u-is-an-m-times-m-o-5

Question

$A$ is an $m 	imes n$ matrix with a singular value decomposition $A=U \Sigma V^{T},$ where $U$ is an $m 	imes m$ orthogonal matrix, $\Sigma$ is an $m 	imes n$ 'diagonal" matrix with $r$ positive entries and no negative entries, and $V$ is an $n 	imes n$ 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 $A .[	ext { Hint: }$ Use Theorem 7 in Section $7.3 . ]$

Jack Chen · Accepted Answer

Okay, so this properly conceived there the men tricks a transpose times a, uh and they&#x27;re supposed that a transposed time. Say, has, um I can barrios Lunda one greet the number two and the wouldn&#x27;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&#x27;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.

Jack Chen · Answer

okay. In this program, we know that a has a single ability composition, utan sigma time speak. So if a is a square metrics, you matrix, um so are on you, sigma and be so off. These three matrices are square Majesty&#x27;s And the way are we up? Know that both you and the V r Also normal mentions is that means the determinant off you equals to pass minus one in this determinant off V close to process minus one in the Since this is a single ability composition that means this Magic sigma we will be a diagonal matrix with off it&#x27;s our singular values Lie on the dagger No entry that tells us the determinant off this kept a sigma will be the protect off this singular values. So in the end, by the property off the determinant determined off a course to the determinant off your times Sigma times being transposed so as you&#x27;re right to be tried Suppose here. But um so this this determinant off three off the product of metrics will be the protect off the determined off each matrix. It would be the determined off you times determinant off capital Sigma times the determined off be transposed in the other. We have plus minus one times Sigma one time Zygmunt to up to six miles on this proof less statements in this program.

Runpeng Li · Answer

foreign problem, mate. Trouble 18. Um, we assume we have a convertible matrix. A where do universities, eh? You first. And we want to find a singular battled the composition of a chambers. So since we have Beatrice Kay which given and we know we can use the singular, better conversation to write A in such a form, it is you. You seem a and e transports. Excuse me. Now we consider the inverse. Okay, so we have you sigma and be transposed. Take the members now through the immersing size that we have transposed chambers and seek a metre seeking my members and you embers now that he&#x27;s dead. Um, Matrix V is orthogonal, so the members will be the same as the transport. So that&#x27;s the That&#x27;s the same as taking the transpose off. Be twice. So he&#x27;s just a baby and seeking my inverse and sees beach excuse also orthogonal. So he&#x27;s just a u transpose now for sigalert Uber&#x27;s now remember the embers of sigma starting the mystery signifies that diagonal matrix so uber is just to take the, um take one over. See my one on diagonal. It&#x27;s the tickets taking the uber self Thea singular values. So this is our Sigma members. This all zeros, all dears. So here is our, ah, single about the composition off you.

Runpeng Li · Answer

cakes for prom 20. We have Hey, she said. Or 12 in all. Or telling a matrix. Now we want to shoot that P Times say, has the same singular values as a now, by our assumption can be decomposed as you times seek my times. U transpose so pee off, eh can be reading There&#x27;s he times you time Sigma times transpose Now Excuse me Now I know what he said since we know peace and on 1000 matrix and you is an orthogonal matrix. So we combine the product off p and you. Then we get a singular value decomposition of P and a p times safe. But at the same time between we observed these singular values that matrix Seema Sigma doesn&#x27;t change. So that means our singular values will be the same. So we&#x27;re done.

Jack Chen · Answer

Okay, so UNICEF program We&#x27;ve already known that the metrics eight has the following decomposition You Sigma Wien transports. So, um, to show that, um to show that to the columns off the argument actors off agents post Wednesday, we can compute h is post A So it just because they you caused to you Sigma the transpose Onda We put analogy Suppose outside times us Sigmund the transports Because to the sick much is suppose you transpose you seem RV transports and that we know that&#x27;s both you and, uh me. Uh, I thought, you know. So the definition off talking a matrix will be you just oppose equals to you, you numerous on the re transpose he was to the interest eso by this condition h is posed a equals two v tons. Six matches Post sigma because you transpose times U equals two And I didn&#x27;t Leah magics. So we can you know this part of the universe Now we move this term the universe to the left that gives us agents pose a V equals toe three times seeking matches pose sigma and we already know that Ah sigma are diagonal matrix So sigma because to something like, Let&#x27;s check in my one have to see my, um And there are some zero terms here. So sick a ma just post time sick. My close to, um something like sick of my wife. Square Sigma to a square. And the sigma is square. Okay, So with this result, we know that if we write out the close to a lot of column vectors, the one I&#x27;ll be too the three up to the end. Then we can decompose this equation as ancients post eight times the I equals two sigma High Square times v i. And this is exactly the ag of the definition of the Eigen vector and again values. So Sigma Chi Square, our, um, again values off a chance post 10 say, and the re eyes. Ah, corresponding Eigen vectors. So this proof on the so if Sigma Square Aiken values off, patron supposed time, say, um, there&#x27;s sticking. My eyes are, ah, singular values off a This proof, the first end of the last statement and full a second statement. We just do it exactly the same thing for a times a transpose. But in this case eight times, they transpose equals Teoh, you sigma The transpose times, the times, stigma chest post temps you And again we used effect that you transport of the transpose equals to the verse. So the protect off this two metrics will be an identity magics. So we can you know this then we have you Sigma Sigma matches post types. You Sorry Decisions. Suppose here. But we can I&#x27;ll rewrite it as your humorous. Then we have a HS post tempts you. You close to a your time signal my time sticking my transposed. So the column vectors off you have you eyes? Are I get better again? Victor&#x27;s off a agent. Suppose so. This proof, the second statement and then we finish on proof here.

Jack Chen · Answer

Okay, so you miss program, we know that young gentlemen tricks, eh? Is positive. Definite. So were we record that, Uh, what&#x27;s the positive definite means? So it means two things. The first wise a transport seacoast way. So ace a symmetric metrics and the the other day is a has I give values. Um, Lunda one up to London Onda uh, I didn&#x27;t values. So Italian vectors Lee one up to the end. Such that a has low following the composition p the p chest post where p equals toward agonal magics with off this Eigen values on the diagonal entrance and the p it&#x27;s a matrix with those again Lecter&#x27;s. So now we&#x27;re back to this, um, problem. This is the information we have. So we consider the metrics a trance post time Say we consider we can see that this matrix and we use this, um, again values and Magan vectors here. So, for example, H is post a times Any Eigen vectors off metrics A We wouldn&#x27;t notice by be high. So I is from one to up to end. I&#x27;m sofrito this since we know that a is the symmetric matrix. So agents post time say you close to eight times eight times V I and we know that the I is an Agon director off a corresponding to wagon value and I So we have eight hands land I the eye sees Lambda is a constant So we have loved I times a v i again the eyes in agony value of a So we have London by square times the high So this tells us the ice Ah, I again vectors off hatreds post time, say, with Eigen value Eigen values Lum dies square eso basically tells us if we want to do the singular value decomposition for matrix A, um that that magics the should be You want each you out to the end, which is executive CMAs the metrics p we are using before. Okay, so now we are for the single ability competition. Rios. We also need to figure out what the metrics you. So by the definition, u equals two. Are you want u I equals to one over sig my eight tons v i but we know that the i r Eigen values argon vectors off magic, say, with Aiken Value Lunda. Hi. So it&#x27;s one over. Sig my lambda High, But what&#x27;s the lamb was listed my here. So we go back to this on this night, we know that the i r again their vectors off a transposed 10 say with Agon Burials Lambda Chi Square. That means the singular value Single Avengers off a are exactly land I So So we have a signal. My costume. Hamdi. So the result will be, um just sorry you should be London tons V I So the result of the V I That means the metrics you will be. You want the chew? Yeah, which is the same as p transposed. Ah, which is the p self? So sorry. It&#x27;s not a piece of its p transpose. Okay, so the singular value decomposition uh, no is just people. So the single ability competition for a can be written as a close to your times? Um, sigma comes the transpose you will be p comes be times p transpose. Oh, it&#x27;s Korean State to the regular regular decomposition for the metric say so we finish our prove here. So this decomposition off us off. Ah, positive. Definite Matrix A is the same as the singular value decomposition fla metrics

Problem

$A$ is an $m \times n$ matrix with a singular val…

Problem 21 Medium Difficulty

Answer

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 $v_{1}$ -

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David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications

Grace H.

Catherine R.

Heather Z.

Maria G.

Absolute Value - Example 1

Absolute Value - Example 2

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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 $v_{1}$ -

Linear Algebra and Its Applications

Grace H.

Catherine R.

Heather Z.

Maria G.

Absolute Value - Example 1

Absolute Value - Example 2

David C. Lay, Steven R. Lay, Judi J. McDonaldLinear Algebra and Its Applications

Grace H.

Catherine R.

Heather Z.

Maria G.

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 $v_{1}$ -

David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications