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Classify the quadratic forms in Exercises $9-18 .$ Then make a change of variable, $\mathbf{x}=P \mathbf{y},$ that transforms the quadratic form into one with no cross-product term. Write the new quadratic form. Construct $P$ using the methods of Section $7.1 .$$$\begin{array}{l}{4 x_{1}^{2}+4 x_{2}^{2}+4 x_{3}^{2}+4 x_{4}^{2}+8 x_{1} x_{2}+8 x_{3} x_{4}-6 x_{1} x_{4}+} \\ {6 x_{2} x_{3}}\end{array}$$

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Algebra

Chapter 7

Symmetric Matrices and Quadratic Forms

Section 2

Quadratic Forms

Introduction to Matrices

Missouri State University

Campbell University

Lectures

01:32

In mathematics, the absolu…

01:11

04:07

Classify the quadratic for…

09:24

03:15

03:01

05:22

02:35

02:33

07:50

05:52

03:42

Make a change of variable,…

we have Thio. Forget it warms that has that has four vegetables. We percent four x one's weird us or X two squared plus four x three square That's for explore, squared and plus ain't X one Thanks wax, too. And eight Next three. Export Next three Sorry, three x four and also negative. Six. Ex wife swore one. That's four and plus six x too extreme. Already down here plus six x two x three. All right. And so, based on this already formed, we can write down a four by four matrix off, eh? Here's a check to diagnose you don't know entries are all force, so it's four or or four and ex wax to has coefficient. Eight. So that means we can take for and four Do the first row, second column and second second Roper's Call him and for extra e x four, and we can start. Start by, um, there's a role for column to have four and corresponding here is for and ex wives for first row. Fourth column will be negative. Three. So here Wilby active three and x two x three. It's the second row. Third column will be three and here is three, and all the other entries are zeros. So here's our four by one matrix. Now, to find the Eigen barrios off these four by formations, it's hard to calculate by hand. So I would recommend just to use program programming skills. Or you can just take this matrix into the orphan Alfa to find out the yeah, Egon betters. So here I'll just write down the four Aiken bettors. So we have a loved one is nine with more police complicity two and London, too. ISS connective one week multiplicity, too. And the corresponding, um again, Victor Ah, we'll be in the check. So be one start with, um, it's kind of confusing. So let's let's just, um it's just to say number one under two. Are you cool? We have nine and or so for the second I can value. Well, let's just say number three and number four to be equal, and they are all negative one, so that is much easier to understand. So that will be number three, because London, for equals negative one look visiting too. All right, since thes these always for Aiken, bettors are not the ah are not the same and Linda, one of the two are positive. And number three number four are negative. So that means quadratic form is indefinite and we can write down. And so right down the, uh we're getting a form under change of miracle. So this will be no, I Why? Once weird Class nine white to square minus. Why? Why three squared minus wife was cleared. And here's our new cold reading for all right. Next thing is thea again vectors. So the Eiken vectors can be found just by, uh, consider the again the definition of bagging Daddy. Wasn't I given vectors? And this gives you take the alumna to this to the system and solve for I can use that really do reduction to find the solution. So, Aiken, Victor, I'll just write down here. And if you're you're interested, you can just, uh, do the role reduction by hand and find out the again vectors one by one, and compare your solutions to mine. And so I'll here. I'll just write down the Eiken values. So the first Aiken value will be in next five naked four and zero and three negative. 40 and three. The second again. Victor will be for by 30 or I three zero. All right, The third Eigen value. Sorry, Agon Vector will be five naked. 403 Bye. Connected. 403 And the last I can vector will be four naked 530 or negative. Five, three, zero. All right, so remember, we need to normalize thes four vectors. So after an organization, uh, yes. Uh, next five square will be 25 next four square would be 16. So this will be one over square 50. So 2050 is equal. Thio five times square, two, five times oil route too. And first I can victories next. Five naked 403 and the same for the rest of Eigen Victor's. So I'll just write down directly won over five times negative square with two and 45 30 And this will be one over five. It's great to. And 456403 and one over five square root too. Or negative. Five, three and zero. All right, so that'll be it on. Did we have therefore Thea Matrix p will be the span off these four vectors, and it would take out one over five Times Square two and inside will be next. Five. Negative four. There were three and 4530 and five make it pours. There are three and the last column or naked 530 Mr by 30 So that's our Matrix.

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