Needed to be solved Part B onlyCorrectly in 15 minutes and get the thumbs up please show neat and clean work for it 3. a) [i) [iii) 1-iv)b) .c) [ªsuch thatLet x = (1, 2) € R² and y = (y₁, y2) € R².Find a symmetric matrix A € M2,2 (R) such thatx Ax= -x + 4x1x₂x².- Find a diagonal matrix D E M2,2 (R) and a rotation matrix Q € M2,2 (R)ii) -1-A~ B.Explain why the equationis an equation of a hyperbola.s] Let up and u₂ be the tangent vectors to the asymptotes of the hyperbolaabove (see fig. 1).x Ax=y Dy, where y = Q¹x.A) Find the canonical (y₁, 92) coordinates of the vectors u₁ and u₂.B)For the same vectors u₁ and u2, find the original (21, 2) coordinates of thevectors up and u₂.-1 Let A, B = Mn (C) be two square matrices.eigenvaluesx Ax = 1Explain what it means that the matrices A and B are similar, i.e.,Prove that if the matrices A and B are similar, then the the the set ofof the matrix A is the same as the set of eigenvalues of the matrix B.Give an example of two matrices A, B E M2,2(C) such thatPROPEUNSA) A is not similar to B; andB) A and B have the same the same set of eigenvalues.Give a detailed justification to your answer.Let a R and letQa =sin a cos asin acos a€ M₂,2 (R)Prove that Qa is orthogonal.Let x € R². Prove that ||Qax| = |x||. Make sure that you carefullystate on the margins of your work, every trigonometric identity and/or any otherresult you use in your proof.

Answered: Image /qna-images/answer/fd9bedf0-024f-4789-9ae0-70fd44694cca.jpg

i) 1 Let A, B E Man (C) be two square matrices. 14 A~ B. Explain what it means that the matrices A and B are similar, i.e., Prove that if the matrices A and B are similar, then the the the set of of the matrix A is the same as the set of eigenvalues of the matrix B. Give an example of two matrices A, B € M2,2(C) such that eigenvalues A) A is not similar to B; and B) A and B have the same the same set of eigenvalues. Give a detailed justification to your answer.

i) 1 Let A, B E Man (C) be two square matrices. 14 A~ B. Explain what it means that the matrices A and B are similar, i.e., Prove that if the matrices A and B are similar, then the the the set of of the matrix A is the same as the set of eigenvalues of the matrix B. Give an example of two matrices A, B € M2,2(C) such that eigenvalues A) A is not similar to B; and B) A and B have the same the same set of eigenvalues. Give a detailed justification to your answer.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter7: Distance And Approximation

Section7.4: The Singular Value Decomposition

Problem 27EQ

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