in-exercises-1-and-2-find-the-change-of-variable-mathbfxp-mathbfy-that-transforms-the-quadratic-fo-2

Question

In Exercises 1 and $2,$ find the change of variable $\mathbf{x}=P \mathbf{y}$ that transforms the quadratic form $\mathbf{x}^{T} A \mathbf{x}$ into $\mathbf{y}^{T} D \mathbf{y}$ as shown.
$$
3 x_{1}^{2}+3 x_{2}^{2}+5 x_{3}^{2}+6 x_{1} x_{2}+2 x_{1} x_{3}+2 x_{2} x_{3}=7 y_{1}^{2}+4 y_{2}^{2}
$$

Runpeng Li · Accepted Answer

Okay, so for problem, too. Excuse me. Um, sorry. So we&#x27;re keeping the quartet performs off x x transpose a Times X. And why transposed the times be at times Why? So we need to find a change of bearable And by the left inside off the give Any question we know our matrix matrix a should be three, three, five honey diagonal and 33 for 1st 1st Rule a versatile 2nd 2nd column because we have the coefficient of X one x two, which is six. So we have three and three for these two entries and same reason we have 111 on the other entries. And because work even were also given the the question on the right hand side, which is the quadratic poor mother off. Uh, why transpose times, times, t times. Why? So that that means our matrix d will be 740 and other terms will be zero. Excuse me. So with that information be given, we know our love. The one will be seven and number two we&#x27;re before on London. Three will be zero. So the next thing is to find out the corresponding wagon vectors. So the idea is to apply our definition off organ vectors and I get values. So we just plugging different wagon values into this room is equality. And find out the vector. Be so I will not go through the process right now because he&#x27;s already the previous married materials. So I&#x27;ll just write down. Yeah, again, Vector&#x27;s first Again, Victor is negative. 0.58 connective, their appointed 58. And that gives every 0.58 this after normalization. And second again, Victor will be negative. 0.41 elective 0.41 point 82. All right, the second. Sorry, the third, Duggan Victor will be going 71.71. So negative points him to want and zero. So our transformation matrix p will span. I&#x27;ll be one b two b three, which is? It is a 0.58 naked 0.8 and 2.58. And the two point What do you want? 2.41 point 82 and 0.71. Negative. 0.71 0 So we&#x27;re done

Jack Chen · Answer

in this program. The symmetric metrics associate to this pathetic phone is a equals 25 to 0 on to a six minus 20 minus 27 So, um, the characteristic equation to this. No tricks. It would be the determinant off a miner&#x27;s love die, which is five miners Lambda Times Six miners loved that times seven minus lambda miners four times seven minus lambda minus four times five months. Lambda um so the characteristic equation you coast through the rope has three solutions Lambda One cause to nylon that week was 26 and 11 3 close to three. So for Lambda One, a miner&#x27;s Lambda one, I equals two R minus 4 to 0 zero minus two minus 20 minus two minus two. So, um, the corresponding I didn&#x27;t backed her will be of the yuan, because to 1/3 12 minus two and the four Lambda to A Minds Lambda to our because to minus one. Choose 0 to 0 minus 20 minus 21 So the corresponding Eigen Vector will be 1/3 time&#x27;s 212 in the four Lambda three in minors. Lambda three. I equals two, 220 to a three minus 20 minus 24 Still, a corresponding Eigen vector will be, um, 1/3 comes minus 2 to 1. So with this wagon veterans, so we can define a magic P equals to 1/3. So use the Sagon directories. Is Colin Space? It&#x27;s one to minus two to want you in the minus two. To what? So with this, um, arrogant, better with this magic speak. If we define the transformation X because to Pete Holmes Why, Where, Why he caused to just why one white You&#x27;re the wife. Three, then x transpose times eight times X equals two um watchers Post times Peter Peaches Post times eight times p times Why in the U will be Why transpose times on 963 Since the multiplication of this matches is, we will be a diagonal matrix times Why so on the result off this pathetic form only contest known Croce Ascension terms Let me is when you have nine. Why one square past six y two square past three wide, three square

Jack Chen · Answer

Okay, so you miss program, we have ah nominal excellence group last 10 x one x two US x two square eso This party nominal corresponds to a car genetic form with the symmetric men shakes 11 and the 55 I&#x27;m 61 x two. So we are looking for We are looking for coordinated change. I do you find you find ah man checks p such that xy cost of p Thomas. Why? Where? Why he closed to just why one and white you. So we are looking for this kind off transformation. So your little figure out was the P we denote this image of mentions by and Toronto find are the item value and Aiken vector off a. This is the real symmetric metric, so we can always figure out these things. So the characteristic equation it will be the determinant off a mine. Islam die. In this case, you will be one minus Lambda Square minus 25 equals zero, which gives us Lambda wine cause the six in the lamb that week was to minus four. So for Lambda One, it was 26 a miner&#x27;s six I equals two minus 55 minus, um, five minus five. So corresponds to and I convict early. One equals 21 over. Route off. Two times 11 We use a normalized again. Victor in the phylum 0.2 equals two minus four a minus minus four I because to oh equals 25555 And they corresponds to another Eigen vector, which is one Have one of the root off to one minus one or we can use minus one. It doesn&#x27;t matter. So the metrics p we are looking for will be a combination off these two column vectors with this, um, Magic p and this, um, transformation. We were have x transpose times eight times. Eggs equals to just plug in this expression we have. Why chills post times p transpose times eight times p times. Why? And, uh, this matrix multiplication gives us, um, a diagonal men tricks. And with Agam Agam variables on the diagonal injuries. So we have my r six and the minus 400 times why? And in the end, away have six y one square proud miners four white to square. This know of crows productor With this transformation

Jack Chen · Answer

Yes. Program. We have a party. Nominal. Two eggs one square minus four x one x two minus x two square which corresponds toe are quadratic form with cemented metrics. A equals two to minus one minus two minus Tokyo. So we are looking for transformation. Ex seacoast of P Hamzawy. Where? Why? Just simply Because too Why one in the white You with this? We want to find out a matrix p such that with this information. Um, the quadratic form contains no, um cross product terms. So in order to figure out this metrics p, we&#x27;re looking for the wagon vectors off. Hey, so the characteristic equation is, um, to mine Islam that times minus one minus Lambda minus four equals zero. Salam DeWine causes three Lambda Tweet coastal minus two. So for Lambda One eight miners, Lunda one, I equals two minus one minus two minus two minus four. Corresponds to Adam that turvey one. You coast to one over. Route five. Tu minus one in the Fulham. That too a minus. Lambda Tau. I cause to four minus two minus 21 corresponds to. And now that I give vector B two week lows toe one overtook five 12 So the matrix p we are looking for we will be one of root of five minus two or minus out to minus one. The 12 So with this transformation, X equals two p times Why this quadratic phone will be x chestful post times a times X because two y chance post times p Just post times a times p times Why? So this matrix multiplication gives us three minus 200 which is a diagonal matrix. And in the end we have three y one square, minus two white. There&#x27;s no uncrossed productor Munir&#x27;s party. No me, no.

Runpeng Li · Answer

given a quadratic form that is a negative X one squared connected X one squared and minus two ex wives too ex one next to minus X two squared. And so again, first thing we need to do is to write out that makes six a which in this case, we checked the politicians or X one squared X two squared. They are both like once of this diagonal will be all naked ones and check the X one x to the Cory vision is connected to so we can separate light headedness has to sum up to negative one. So that&#x27;s negative. One negative one. So here&#x27;s our matrix. And, um, the next thing is to find out the wagon betters So do that. We used a determinant a minus Lambda chi. So for this matrix, it is just next one minus number. Uh, next one next one next one month Dum dum and we&#x27;re looking for the determined. So this will be no one Minds Lunda squared minus one is zero. So this means lambda iss Okay, um, the students that step by step so we know by the previous expression we know that Lambda plus One the best one squared is one. So that means loved a plus one is one. So we can take number one to be elected to. Or London too to be C zero. Yeah. So that means since we have, ah, negative value value and we also have the there I can value. So that means this matrix a has the court verdict form indefinite. Okay. And now we can write down Thea quadratic forms off the change of variables. That is, um, next to active to why one squared? Why one square? Okay. And so we take the take the Eigen value back to our matrix. So that is to consider the system a axe equals London acts. So by the first row, we have negative X one minus x two iss Here we first apply. Our first I could value, which is connected to is naked too. Oh, excellent. So what? This means it implies that x one. His ex, too, is equal to X to all right. And for the second leg of value, we have x one minus x two equal Cyril. So that means X one is negative x two. So the first idea in vector we have is 11 Well, this second I in Vector we have is one negative one. So in order to normalize it divided it. We divide this by square it up too. So this is the one over square with 21 one. And same for the second dog in Victor one. Negative. Okay, so, Bye. So the matrix p we&#x27;ll just be this band, uh, these two Aiken, Victor&#x27;s. So we have one over square root too. That&#x27;s a scaler. Kinds The matrix. 111 Next one. So that&#x27;s it.

Steven Bridges · Answer

But this question were again finding new idea Jax, which is the derivative. So for each one of these, we&#x27;re going to use the power rule, which is to take the exponents, bring it in front as a constant, and then lower the power by one. So what this is going to do for us here is we&#x27;re going to end up with three and we&#x27;ll have our X over three. And the new power goes down too. Plus two X over too. In the new Power one and the derivative of X is one. So we simplify this. We end up with our C Y. D X is equal to X squared close axe plus one.

Problem

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

Problem 2 Easy Difficulty

In Exercises 1 and $2,$ find the change of variable $\mathbf{x}=P \mathbf{y}$ that transforms the quadratic form $\mathbf{x}^{T} A \mathbf{x}$ into $\mathbf{y}^{T} D \mathbf{y}$ as shown.
$$
3 x_{1}^{2}+3 x_{2}^{2}+5 x_{3}^{2}+6 x_{1} x_{2}+2 x_{1} x_{3}+2 x_{2} x_{3}=7 y_{1}^{2}+4 y_{2}^{2}
$$

Answer

$P=\left(\begin{array}{lll}{\mathbf{u}_{1}} & {\mathbf{u}_{2}} & {\mathbf{u}_{3}}\end{array}\right)$

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

Linear Algebra and Its Applications

Catherine R.

Caleb E.

Kristen K.

Michael J.

Absolute Value - Example 1

Absolute Value - Example 2

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$P=\left(\begin{array}{lll}{\mathbf{u}_{1}} & {\mathbf{u}_{2}} & {\mathbf{u}_{3}}\end{array}\right)$

Linear Algebra and Its Applications

Catherine R.

Caleb E.

Kristen K.

Michael J.

Absolute Value - Example 1

Absolute Value - Example 2

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

Catherine R.

Caleb E.

Kristen K.

Michael J.

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

Linear Algebra and Its Applications