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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 .$$$x_{1}^{2}-6 x_{1} x_{2}+9 x_{2}^{2}$$

$Q(\mathbf{y})=10 y_{1}^{2}$

Algebra

Chapter 7

Symmetric Matrices and Quadratic Forms

Section 2

Quadratic Forms

Introduction to Matrices

Campbell University

Baylor University

Lectures

01:32

In mathematics, the absolu…

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07:50

Classify the quadratic for…

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Make a change of variable,…

Okay, so Eunice proper and we have a phenomenal X one square, minus six x Y x two plus nine x two square. So there is a cross product turn here on this party naming, of course, bones to a quadratic form with man checks a equals toe minus three minus three nights. So we want to find out a transformation such that there's no on cross pro doctor ERM under this transformation. So we're looking for the Agon value and I backed off magic say the characteristic equation is a determined off a mine. Islam die cost zero. Which is why mine, Islam, that Pam's nine miners Lambda minus 90 causes Europe. Salam DeWine causes zero in the lamb that we close to 10 for Lambda one Any miners? Lambda I I'm no one times I equals to yourself. So the corresponding Eigen vector will be won over route off 10 times 31 And the lamb that week was to turn the corresponding I connect her. It was the one of a root of 10 one minus three. So the metrics p we're looking for will be won over route off 10 31 one, minus three. So with this transformation, X equals two p times white. And why just equals Teoh? Why one times y one y two. So with this transformation, this quadratic form equals to, um why transpose times peaches posed as a times p times Why? So this matrix multiplication we will be and a diagonal magics with dragon value ozone diagonal entrance. So we have 10 times y to a square. There's no cross pro doctor amused expression.

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Classify the quadratic forms in Exercises $9-18 .$ Then make a change of var…

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