Search All of the Math Forum:
Views expressed in these public forums are not endorsed by
Drexel University or The Math Forum.
|
|
|
|
Re: unitary matrix
Posted:
May 18, 2006 4:20 PM
|
|
"katy" <mcld@mega.ist.utl.pt> wrote in message
news:1147894859.922823.226600@u72g2000cwu.googlegroups.com...
> Hi!
>
> I'm trying to prove that when A and B are positive definite (Det >0)
>
> U= (A^-1/2 B A^-1/2)^1/2 A^1/2 B^-1/2 is an unitary matrix
>
> I simplified U:
>
> U= (A^-1/2 B A^-1/2)^1/2 A^1/2 B^-1/2
>
> = A^-1/4 B^1/2 A^-1/4 A^1/2 B^ -1/2
> = A^-1/4 B^1/2 A^1/4 B^-1/2
>
> I calculated U^T:
>
> U^T = [ (A^-1/2 B A^-1/2)^1/2 A^1/2 B^-1/2 ]^T=
>
> = (B^-1/2)^T (A^1/2 )^T [( A^-1/2 B A^-1/2)^1/2]^T
>
> = (B^-1/2)^T (A^1/2 )^T ( A^-1/4 B^1/2 A^-1/4)^T
>
> = (B^-1/2)^T (A^1/2 )^T ( A^-1/4) ^T (B^1/2)^T (A^-1/4)^T
>
> = (B^-1/2)^T (A^(1/2-1/4)) ^T (B^1/2)^T (A^-1/4)^T
>
> = (B^-1/2)^T (A^1/4) ^T (B^1/2)^T (A^-1/4)^T
>
>
> I also calculated U^-1:
>
> U^-1= [ (A^-1/2 B A^-1/2)^1/2 A^1/2 B^-1/2 ]^-1
>
> = B^1/2 A^-1/2 ( A^-1/2 B A^-1/2)^-1/2
>
> = B^1/2 A^-1/2 A^1/4 B^-1/2 A^1/4
>
> = B^1/2 A^-1/4 B^-1/2 A^1/4
>
> But i didn't achieved any conclusion :(
>
>
>
> If the space is real I've proved that U is unitary (using the fact that
> A*= A^T, so A^-1 = A^T)
>
> So i verified that
>
> U U^T= I
>
>
> [A^-1/4 B^1/2 A^1/4 B^-1/2 ] [ (B^-1/2)^T (A^1/4) ^T (B^1/2)^T
> (A^-1/4)^T]= I =
>
> <=> A^-1/4 B^1/2 A^1/4 B^-1/2 (B^-1/2)^T (A^1/4) ^T (B^1/2)^T
> (A^-1/4)^T = i
>
> <=> A^-1/4 B^1/2 A^1/4 B^-1/2 (B^1/2) (A^1/4) ^T (B^1/2)^T
> (A^-1/4)^T = i
>
> <=> A^-1/4 B^1/2 A^1/4 (A^1/4) ^T (B^1/2)^T (A^-1/4)^T = I
>
> <=> A^-1/4 B^1/2 A^1/4 (A^-1/4) (B^1/2)^T (A^-1/4)^T = I
>
> <=>A^-1/4 B^1/2 (B^1/2)^T (A^-1/4)^T = I
>
> <=> A^-1/4 B^1/2 (B^-1/2) (A^-1/4)^T = I
>
> <=> A^-1/4 (A^-1/4)^T = I
>
> <=> A^-1/4 (A^1/4) = I
>
> <=> I=I
>
>
>
>
> Can somebody help me to prove that U is unitary in any case?
>
>
> Thank you very much,
>
> Catarina Dias
>
You may find this easier to do if you don't simplify U. Just compute U*U^T.
________________________________
Eric J. Wingler (wingler@math.ysu.edu)
Dept. of Mathematics and Statistics
Youngstown State University
One University Plaza
Youngstown, OH 44555-0001
330-941-1817
|
|
|
|
