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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) = det (B). Give an example to show that the converse does not hold. d) Show that there are at least n equivalences classes of the relation ~that consist of matrices with determinant 0.

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) = det (B). Give an example to show that the converse does not hold. d) Show that there are at least n equivalences classes of the relation ~that consist of matrices with determinant 0.

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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Definition: If A, B E Matnxn (C), we say A and B are conjugate if there is an invertible matrix CE Matnxn (C) such that C-¹ AC = 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 PÂ(t) = PÂ(t), that is, A and B have the same characteristic polynomial. c) If A and B are conjugate, prove det(A) = det (B). Give an example to show that the converse does not hold. d) Show that there are at least n equivalences classes of the relation~ that consist of matrices with determinant 0. (Hint: You may use part (b) even if you did not solve it.)
Transcribed Image Text:Definition: If A, B E Matnxn (C), we say A and B are conjugate if there is an invertible matrix CE Matnxn (C) such that C-¹ AC = 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 PÂ(t) = PÂ(t), that is, A and B have the same characteristic polynomial. c) If A and B are conjugate, prove det(A) = det (B). Give an example to show that the converse does not hold. d) Show that there are at least n equivalences classes of the relation~ that consist of matrices with determinant 0. (Hint: You may use part (b) even if you did not solve it.)
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