in-exercises-3-6-find-a-the-maximum-value-of-qx-subject-to-the-constraint-xt-x1b-a-unit-vector-u-whe

Question

In Exercises $3-6,$ find (a) the maximum value of $Q(x)$ subject to the constraint $x^{T} x=1,(b)$ a unit vector $u$ where this maximum is attained, and $(c)$ the maximum of $Q(x)$ subject to the constraints $\mathbf{x}^{T} \mathbf{x}=1$ and $\mathbf{x}^{T} \mathbf{u}=0$
$$
\begin{array}{l}{Q(\mathbf{x})=5 x_{1}^{2}+6 x_{2}^{2}+7 x_{3}^{2}+4 x_{1} x_{2}-4 x_{2} x_{3}} \ {	ext { (See Exercise } 1 . )}\end{array}
$$

Jack Chen · Accepted Answer

so by excess one, Um, the maximum value off this quadratic a phone Q X, um, subject to the constraint, um, X transpose X equals to one Lemonis X is a unit vector. I will be the makes him all Eigen value. So in this case, the Mexi Magan Value is loved. Ah wai In close to nine, that means this is the next Mama Leo off q X subject Teoh any you meet back to Rex and ah one q X equals tonight. So when q x a chief&#x27;s its maximum value, the corresponding that unit vector will be the item backed her, um, associates to this aggravator of nine. So it&#x27;s 1/3 um, 12 miners toe and the CCT since this is ah quadratic terms. So, um, the choice off this Eigen vector could be plows miners. So we have two different choices. No, If we add another constraint, that&#x27;s, um, X transpose times X equals zero. Say so you need vector, um, and the X transpose times you it causes your That means X is perpendicular to on this unions factor. Then the maximum value off this chaotic time, Q X. We will be the second largest agon. Valeo now belong that we caused to six

Runpeng Li · Answer

Okay, So for problem for we could given order to perform, we need to find out the and make some money off. Off the Q axe subject to constrained X transpose times X equals zero. It was one. So that&#x27;s the first part. We first conceded. The first part are a So we write down the corresponding matrix A we thank you been called reading form. We have 335 on the diagonal and three, 31111 This is just our matrix from from our exercise too. So Okay, the next thing is to find out the argon values. Um, since we were that worried, we&#x27;ve already down exercise to so I&#x27;ll just write down the I convey to stoop directly just seven, four and zero. So our, um maximum off cue iss can be achieved when we take the vector to be. Yeah, I can value corresponding to the Eiken vector. I can vector corresponding to die, give value of seven. She&#x27;s the one right down the values off the one which is 90.58 negative 0.58 and naked 0.58. So we just plucking the value off the rails off V one and we get the maximum value, which is, uh, which is close to seven. I should be able to London one. Okay, R b. So we need Thio. Find a unit vector where? Anti Mexican. He&#x27;s attained it. So the union victory is just our again, Victor, just like what we did before in part a just a b one. So our unit vector will be you equal to be one. Okay, so for our c So two to work out hard. See, we need to apply overthere. Um, in our textbook Excuse me, theorem seven. Which says if we wear given the constraints, X transpose times x you to be one and extra syllables. X transpose times U is zero than then. The maximum is London too. And it is achieved by, um it achieved by the corresponding again. Victor say maximum well, peachy will be achieved by the second I get Muktar, which in our case, should be, um que Max. Uh, connective 0.41 active 0.41. And coined 82. So that&#x27;s it.

Runpeng Li · Answer

Okay, so four problem six. Um, based on our given quadratic form, we can write down the Matrix. A. This case should be a two by two matrix 39 antagonist and four or for the rest entries. And by calculating the idea values wishes tonight and 11. So I want to live. Okay. 1 11 now to attain to attain the maximum maximum value l Q AG subsidies. Ex subject to the constraint. Next X transpose times actually was one that is, too. To find out the largest aiken value, I get better first. So are our largest. I can very well be 11 which is stick. She&#x27;s obtained by the XY moment. Uh, the cory&#x27;s on the again. Victor, be too now, course money by Victor. I&#x27;ll just, uh, write it down because it&#x27;s a two by two matrix. You just need to consider the definition here a times B, post number, times fee, and you&#x27;re done. So, Thea, again Victor will be negative 0.45 next to Pointe 89. So here we can take the unit vector to be equal to our Vector two and we&#x27;re done now. So for Parsi So we are given the constraint X transpose times acts he was one and x transpose times of U equals zero so that these two find a second largest Agnes hugging battle, which is one in this case we only have two activators loved one will be one which is attained buying the maximum of be one where everyone is, say, negative point 89 point or if I so we&#x27;re done.

Jack Chen · Answer

So before we solving this problem, when first we need to figure out the Eigen value and I connectors off the matrix, a associate, this cultural form. So the metric saying for this Kojak a phone will be one minus five minus 51 So, in order to find the dragon values, we need to self this characteristic equation the determined off a minus lambda I equals to one. Mine is Lambda Square, minus 25 equals zero. So we have to Agam values London wankers. Second run that week was to minus four. So part a, the maximum more value off Q x on even eso for any you need to make her kicks. So the answer to this problem will be six because this is the largest. I get value off magic sake, and then you have to be. We want to figure out what kind off you need to factor. Which union? Vector X Um we are chief, this our maximum value. So we need to fit so that the answer is the dragon factory associated wave. So for the Adamek a minus six, I equals two, um minus five, minus five minus five minus five. Still of course, bone The united vector U equals to one over root of two. Ah, Why minus one. So when X equals two plus minus one over route off to on one minus one. Que eggs achieve seats in Mexico. Value in. See if we add another constraint. Exchange post times you because toe one onions. We&#x27;re looking for a unit vector perpendicular to this you we found before. So you know this case that maximum value off q X we will be the second largest Eigen value that will be spinal for

Runpeng Li · Answer

Okay, So in this problem, we need to find the Mexican value off acute acts without, um without finding the vector where the Mexican he&#x27;s attained. So this is actually we need to apply our, um, our knowledge and Nash bro. So at first, the first, the first thing we need to remark here is the inequality X squared plus x two square speaker or equal to two excellent x two. So that means our to X one x two will be smaller or equal to one. And so our ex likes to will be smaller or equal to 1/2. Now recall her or assumption of q X, which is connective three x one square. That&#x27;s five x two squared minus two x one x two So minus two x y excuse. We just apply this inequality times two times Negative, too. 22 Ex LAX, too. It&#x27;s bigger. We could to one negative one. So what that means this expression will be bigger. We put too negative. Three x one squared plus five excuse Weird. Minus one. Okay. And now we can also take X one squared to be equal to one minus x two square. It&#x27;s a spiral assumption. So we have negative three one minus x two squared 1st 5 X two squared minus one. So that is active three US three excuse squared US five x two squared my one. So that is aid X two squared minus four. So no, still to make the to make you accept as large as possible So we can check that the lower bound off co ax, which is eight x two squared minus four So we can&#x27;t check how large this lower bound can be. So that means we need to take X two s started. It&#x27;s possible the biggest X two is one and x two cannot be feared or what? Because if that extreme is bigger than one than our our assumption next one squared plus X two squared, it was one will be filled. So the largest X do we can take just one. So at this moment, X one will be easy so that I hear the expression here will be equal to, uh, four. Now, we just plugging the set of values are facts into our, um we&#x27;re ready for So we have Q um, que axe Hope? Sorry. Q. Max is equal to now x Y zero. So this term will be zero X two is one. So here will be five and X Y zero. So these two will be zero. So we have fine off course 55 speaker or two for so that is a Mexican valley we can attain.

Jack Chen · Answer

she&#x27;s so, uh, given this quadratic form the next video off Q x ah subject two X transpose times X equals to one is, um the maximal I am value off the metrics A. So, in this case, a close to seven minus one minus 13 So the agon value off a is the solution to this characteristic equation that would be seven minus lambda times three minus lambda minus one equals zero. So Lamda one equals 25 plus Ruutel five and the Lambda Chewy, close to five minus root. If I so sees Lambda Wise Greater Lambda to sell a maximum value equals 25 plus Ruto five.

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In Exercises $3-6,$ find (a) the maximum value of…

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

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

Linear Algebra and Its Applications