Is there a way to show that Ak is unitarily similar to Bk using induction?

[chuy52506]

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 A^k is unitarily similar to B^k 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?

 

[vela]

You want $A^k = U^\dagger B^k U$. You don't need to take U and its adjoint to the k-th power.

 

[chuy52506]

so then would it be simply
Aⁿ⁺¹=U^*Bⁿ⁺¹U?? and I am finished?

 

[vela]

Well, you need to prove that statement is true.

 

[chuy52506]

so i would have
AⁿA=U^*BⁿBU

 

[chuy52506]

i am stuck??=[

 

[vela]

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.