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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.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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.
Transcribed Image Text: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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