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Topic: unitary matrix
Replies: 2   Last Post: May 18, 2006 5:58 PM

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katy

Posts: 13
Registered: 11/2/05
unitary matrix
Posted: May 17, 2006 3:41 PM
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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




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