- #1
RJLiberator
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Homework Statement
Let |v> ∈ ℂ^2 and |w> = A|v> where A is an nxn unitary matrix. Show that <v|v> = <w|w>.
Homework Equations
* = complex conjugate
† = hermitian conjugate
The Attempt at a Solution
Start: <v|v> = <w|w>
Use definition of w
<v|v>=<A|v>A|v>>
Here's the interesting part
Using properties of a complex inner product, we can take out the unitary matrix A on the right hand side. When we do so, what do we get?
Is it A*A or is it A†A ?
If it is A†A this proof is complete, as sine A is unitary, this means A†A = the identity matrix and we get equality.
If the definition is A*A then we have to do more work.
I am getting mixed up, as I'm seeing both definitions floating around and with the abusive notation everywhere it is making me second guess which definition is right.
Is there any clarity on this issue here?