Unitarily similar matrices

  • Thread starter chuy52506
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  • #1
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Homework Statement


SUppose A and B are nxn matrices in the complex field and that A is unitarily similar to B.

Homework Equations


Show that Ak is unitarily similar to Bk for all k=1,2,3,..


The Attempt at a Solution


I used induction to show its true for k=1 which it is.
Then for k=n+1,
An+1=(U*)n+1Bn+1Un+1
AnA=(Un)*(U*)B(Bn)UnU.

That is as far as i got, any help?
 

Answers and Replies

  • #2
vela
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You want [itex]A^k = U^\dagger B^k U[/itex]. You don't need to take U and its adjoint to the k-th power.
 
  • #3
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so then would it be simply
An+1=U*Bn+1U?? and im finished?
 
  • #4
vela
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Well, you need to prove that statement is true.
 
  • #5
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so i would have
AnA=U*BnBU
 
  • #6
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i am stuck??=[
 
  • #7
vela
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so i would have
AnA=U*BnBU
Nope, you have, using the induction hypothesis An=U*BnU,

An+1 = AnA = (U*BnU)A

Now write that last factor of A in terms of U and B.
 

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