Please provide detailed solution for III part asap to get a upvote 3. а)Let x = (1.12) e R² and y =(1. 2) E R².i) Find a symmetric matrix A E M22(R) such thatx" Ax = r + 4ry + y*.ii) Find a diagonal matrix De M22(R) and a rotation matrix Q € M2.2(R) such thatx' Ax = y" Dy, where y = Q"x.iii) Explain why the equationx" Ax = 1is an equation of a hyperbola.iv) Find the canonical (yı, 92) coordinates of the tangent vectors to the asymptotes ofthe hyperbola above. That is, find vectors uj and uz as shown on the fig 1.v) Find the original (1.12) coordinates of the tangent vectors to the asymptotes ofthe hyperbola above. That is, find vectors uj and uz as shown on the fig 1.

Answered: Image /qna-images/answer/fc8f3f89-fa71-49d8-b8ec-031d8de3ba35.jpg

Let x = (r1.12) E R? and y = (y.2) E R². i) Find a symmetric matrix A e M2.2(R) such that x" Ax = 1 + 4ry + y*. %3D i) Find a diagonal matrix De M22(R) and a rotation matrix QE M22(R) such that x Ax = y" Dy, where y = Q"x. 5) Explain why the equation x' Ax = 1 is an equation of a hyperbola.

Let x = (r1.12) E R? and y = (y.2) E R². i) Find a symmetric matrix A e M2.2(R) such that x" Ax = 1 + 4ry + y*. %3D i) Find a diagonal matrix De M22(R) and a rotation matrix QE M22(R) such that x Ax = y" Dy, where y = Q"x. 5) Explain why the equation x' Ax = 1 is an equation of a hyperbola.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter5: Orthogonality

Section5.4: Orthogonal Diagonalization Of Symmetric Matrices

Problem 27EQ

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Please provide detailed solution for III part asap to get a upvote

3. а)
Let x = (1.12) e R² and y =
(1. 2) E R².
i) Find a symmetric matrix A E M22(R) such that
x" Ax = r + 4ry + y*.
ii) Find a diagonal matrix De M22(R) and a rotation matrix Q € M2.2(R) such that
x' Ax = y" Dy, where y = Q"x.
iii) Explain why the equation
x" Ax = 1
is an equation of a hyperbola.
iv) Find the canonical (yı, 92) coordinates of the tangent vectors to the asymptotes of
the hyperbola above. That is, find vectors uj and uz as shown on the fig 1.
v) Find the original (1.12) coordinates of the tangent vectors to the asymptotes of
the hyperbola above. That is, find vectors uj and uz as shown on the fig 1.

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