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Make a change of variable, $\mathbf{x}=P \mathbf{y},$ that transforms the quadratic form $x_{1}^{2}+10 x_{1} x_{2}+x_{2}^{2}$ into a quadratic form with no cross-product term. Give $P$ and the new quadratic form.

$6 y_{1}^{2}-4 y_{2}^{2}$

Algebra

Chapter 7

Symmetric Matrices and Quadratic Forms

Section 2

Quadratic Forms

Introduction to Matrices

McMaster University

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

Lectures

01:32

In mathematics, the absolu…

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03:24

In Exercises 1 and $2,$ fi…

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05:52

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07:50

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05:22

04:07

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'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'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'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

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