# Unitarily similar matrices

## 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?

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

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

vela
Staff Emeritus
Homework Helper
Well, you need to prove that statement is true.

so i would have
AnA=U*BnBU

i am stuck??=[

vela
Staff Emeritus
Homework Helper