- #1
ognik
- 643
- 2
A particular similarity transformation yields:
A' = UAU-1
(A')\(\displaystyle \dagger\) = UA\(\displaystyle \dagger\)U-1
if the adjoint relationship is preserved (A'\(\displaystyle \dagger\)=A\(\displaystyle \dagger\)') and det(U)=1, then Show that U must Be Unitary
-----------------------------
I think I've nearly got it as follows, but not sure my arguments are completely valid:
|det(U)|=1 is one of the conditions for U to be Unitary
From A' = UAU-1:
$ ({A}^{'})^{\dagger} = (UA{U}^{-1})^\dagger = (A{U}^{-1})^\dagger{U}^{\dagger} = ({U}^{-1})^\dagger {A}^{\dagger} {U}^{\dagger} $
So, from (A')\(\displaystyle \dagger\) = UA\(\displaystyle \dagger\)U-1 above:
$ ({U}^{-1})^\dagger{A}^{\dagger}{U}^{\dagger} = U{A}^{\dagger}{U}^{-1} $
Then equating the RH elements, $ {U}^{\dagger}={U}^{-1} $ which makes U unitary...provided we can also show the the LH elements are equal, IE show $ ({U}^{-1})^\dagger = U $
Now if U is Unitary, then $ ({U}^{-1})^\dagger = ({U}^\dagger)^\dagger = U $
This last step I don't think I have argued correctly, how would I improve the argument, or is there a better way of doing this proof? Thanks
A' = UAU-1
(A')\(\displaystyle \dagger\) = UA\(\displaystyle \dagger\)U-1
if the adjoint relationship is preserved (A'\(\displaystyle \dagger\)=A\(\displaystyle \dagger\)') and det(U)=1, then Show that U must Be Unitary
-----------------------------
I think I've nearly got it as follows, but not sure my arguments are completely valid:
|det(U)|=1 is one of the conditions for U to be Unitary
From A' = UAU-1:
$ ({A}^{'})^{\dagger} = (UA{U}^{-1})^\dagger = (A{U}^{-1})^\dagger{U}^{\dagger} = ({U}^{-1})^\dagger {A}^{\dagger} {U}^{\dagger} $
So, from (A')\(\displaystyle \dagger\) = UA\(\displaystyle \dagger\)U-1 above:
$ ({U}^{-1})^\dagger{A}^{\dagger}{U}^{\dagger} = U{A}^{\dagger}{U}^{-1} $
Then equating the RH elements, $ {U}^{\dagger}={U}^{-1} $ which makes U unitary...provided we can also show the the LH elements are equal, IE show $ ({U}^{-1})^\dagger = U $
Now if U is Unitary, then $ ({U}^{-1})^\dagger = ({U}^\dagger)^\dagger = U $
This last step I don't think I have argued correctly, how would I improve the argument, or is there a better way of doing this proof? Thanks