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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 .$$$4 x_{1}^{2}-4 x_{1} x_{2}+4 x_{2}^{2}$$

Hence the required new quadratic form is $Q(y)=2 y_{1}^{2}+6 y_{2}^{2}$

Algebra

Chapter 7

Symmetric Matrices and Quadratic Forms

Section 2

Quadratic Forms

Introduction to Matrices

Campbell University

McMaster University

Harvey Mudd College

University of Michigan - Ann Arbor

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In mathematics, the absolu…

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Make a change of variable,…

Okay, so in this question, we are given a phenomenal So this pollen nominal corresponds toe quadratic form with a symmetric matrix A equals two for minus two minus 24 And they were looking for information on turned this quadratic form into one with without any cross product room. So it basically want to eliminate this one. So we are looking for We are looking for metrics p such that X equals two Pete Holmes. Why? Where? Why equals two? Ah y one y two. So in order to figure out this metrics P, we need to figure out the agon value in Agra vector for the metric say so. The characteristic equation is their top. A minor slammed I, the determined off a equals 20 off a minus lambda lambda because zero So we have four miners. Lambda Square minus four equals zero. So we have to wagon May tomatoes. Lambda wine caused 26 and the lamb that we close to two for Lambda One a miner's Lambda one I because to minus two minus two minus two minus still corresponding to Agon. Vector B one equals 21 over it. Off to one minus one in the lump that we coasted to a mine Islam, that why equals two Tu minus two minus 22 corresponding toe a second Eigen vector. You'll be one of the root off two times 11 So the make six we're looking for P equals two one of Rudolf two one minus one. The 11 So with this transformation X transpose times a times X equals to white transpose times p transpose times eight times p times Why, which becomes so the this matrix multiplication gives us, um 62 which is a diagonal magics and Indiana. We have six of why one square pressed too wide to square. Also, this, um, party nominee opened as o no across product er's.

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In mathematics, the absolute value or modulus |x| of a real number x is its …

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

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