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

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Robert Israel

Posts: 11,902
Registered: 12/6/04
Re: unitary matrix
Posted: May 18, 2006 5:58 PM
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In article <1147894859.922823.226600@u72g2000cwu.googlegroups.com>,
katy <mcld@mega.ist.utl.pt> wrote:
>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

There's your first mistake. The square root of a product of
matrices is not necessarily the product of the square roots when the
matrices don't commute.

As Eric Wingler suggests, this "simplification" is not needed.
However, it may help (after writing out U U^T) to substitute
B = A^(1/2) C A^(1/2).

Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada

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