Thread: Proof that unitary operator is inner product preserving.

Having read the proof for this from http://www-inst.eecs.berkeley.edu/~c...s/lecture4.pdf, I am struggling to understand exactly where the first and second steps come from. From which property of the inner product does it follow that <A|B> = Adjoint(A)B ? Thanks in advance.

Hey StaryNight.

I remember a while ago that it has to do with the commutative aspect of the inner product and then this makes the object Hermitian or at least having the transpose equal to the original.

Originally Posted by chiro

Hey StaryNight.

I remember a while ago that it has to do with the commutative aspect of the inner product and then this makes the object Hermitian or at least having the transpose equal to the original.

Hi chiro, could you please elaborate? Thaks.

Basically, it has to do with inner product spaces:

Inner product space - Wikipedia, the free encyclopedia

Spectral theorem - Wikipedia, the free encyclopedia

i will use U* to denote the conjugate-transpose or adjoint of U.

by definition, this is the linear transformation for which: <Uv,w> = <v,U*w> for any vectors (or "kets" as physicists like to call them) v,w.

if U is unitary, which means U*U = UU* = I (the identity transformation), then using Uw in place of w gives:

<Uv,Uw> = <v,U*(Uw)> = <v,(U*U)w> = <v,Iw> = <v,w>

note that in a complex inner-product space, <v,_> is often defined as the linear functional v*(_), that is: <v|w> = (|v>)*|w> (a row consisting of the conjugates of the coordinates of v times a column consisting of the coordinates of w).

this insures sesquilinearity (most mathematical treatments do the opposite and define <v,w> = (v*w)*, in other words we have linearity in the first argument, rather than the second. this is your typical "left-handed" versus "right-handed" choice...that is, in physics we have: <x|y> = <y,x> according to the "usual" mathematical definition).

Brilliant, thanks, I didn't realise that this was the definition of the adjoint.

Originally Posted by Deveno

i will use U* to denote the conjugate-transpose or adjoint of U.

by definition, this is the linear transformation for which: <Uv,w> = <v,U*w> for any vectors (or "kets" as physicists like to call them) v,w.

if U is unitary, which means U*U = UU* = I (the identity transformation), then using Uw in place of w gives:

<Uv,Uw> = <v,U*(Uw)> = <v,(U*U)w> = <v,Iw> = <v,w>

note that in a complex inner-product space, <v,_> is often defined as the linear functional v*(_), that is: <v|w> = (|v>)*|w> (a row consisting of the conjugates of the coordinates of v times a column consisting of the coordinates of w).

this insures sesquilinearity (most mathematical treatments do the opposite and define <v,w> = (v*w)*, in other words we have linearity in the first argument, rather than the second. this is your typical "left-handed" versus "right-handed" choice...that is, in physics we have: <x|y> = <y,x> according to the "usual" mathematical definition).