Question

Show that unitary transformations of vectors preserve scalar products. (complex vectors/matrices problem)

Answer 1

U preserves inner products, i.e. (v, w) = (Uv,Uw)

U^?t = conjugate-transpose or adjoint of U.

By definition, this is the linear transformation for which: <Uv,w> = <v,U^?t_?w> for any vector v,w.

If U is unitary, which means U^?tU = UU^?t = I (the identity transformation), then using Uw in place of w gives:

<Uv,Uw> = <v,U^?t(Uw)> = <v,(U^?tU)w> = <v,Iw> = <v,w>

Note that in a complex inner-product space, <v,_> is often defined as the linear functional v^?t(_),

that is: <v|w> = (|v>)^?t|w> (a row consisting of the conjugates of the coordinates of v times a column consisting of the coordinates of w).