Definition: If A, B = Matnxn (C), we say A and B are conjugate if there is an invertible matrix C Matnxn (C) such that C-¹AC = B. 3. a) Prove that the relation~ defined by A~ B if and only if A and B are conjugate is an equivalence relation on the set Matnxn (C). b) If A and B are conjugate, prove that PA (t) = PB (t), that is, A and B have the same characteristic polynomial. = c) If A and B are conjugate, prove det(A) does not hold. det (B). Give an example to show that the converse

Definition: If A, B = Matnxn (C), we say A and B are conjugate if there is an invertible matrix C Matnxn (C) such that C-¹AC = B. 3. a) Prove that the relation~ defined by A~ B if and only if A and B are conjugate is an equivalence relation on the set Matnxn (C). b) If A and B are conjugate, prove that PA (t) = PB (t), that is, A and B have the same characteristic polynomial. = c) If A and B are conjugate, prove det(A) does not hold. det (B). Give an example to show that the converse

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter3: Matrices

Section3.7: Applications

Problem 76EQ

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