In article <email@example.com>, katy <firstname.lastname@example.org> wrote: >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
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).