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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 .$$$-x_{1}^{2}-2 x_{1} x_{2}-x_{2}^{2}$$

Hence the required new quadratic form is $Q(y)=-2 y_{1}^{2}$

Algebra

Chapter 7

Symmetric Matrices and Quadratic Forms

Section 2

Quadratic Forms

Introduction to Matrices

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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's negative. One negative one. So here'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'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'll just be this band, uh, these two Aiken, Victor's. So we have one over square root too. That's a scaler. Kinds The matrix. 111 Next one. So that's it.

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