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-¹AC = B. 3. a) Prove that the relation defined by AB 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 P₁(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. (Hint: You may use part (b) even if you did not solve it.)
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-¹AC = B. 3. a) Prove that the relation defined by AB 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 P₁(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. (Hint: You may use part (b) even if you did not solve it.)
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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d
Transcribed Image Text: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)
det (B). Give an example to show that the converse
does not hold.
that consist of matrices with
d) Show that there are at least n equivalences classes of the relation
determinant 0.
(Hint: You may use part (b) even if you did not solve it.)
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