Let x be a nx1 vector in Cn and y = U*x is another vector in Cn, where U is a unitary matrix and * represents the conjugate transpose.
The Euclidean norm of y equals 1 if and only if the Euclidean norm of x equals 1:
||y|| = 1 if and only if ||x|| = 1.
I can't prove this statement and don't know where to start!
Thanks so much!
Thanks a lot guys. 2 different proofs, but both very nice.
Every matrix has at least one eigenvalue (fundamental theorem of algebra)
If U is unitary then U* is unitary too
Every eigenvalue of a unitary matrix has absolute value of 1
So: ||y|| = ||U*x|| = ||lambda x|| = |lambda| · ||x|| = ||x||
If y = U*x then x = Uy
So: ||x||^2 = x*x = y*U*Uy = y*y = ||y||^2 <=> ||x|| = ||y||
Thanks a lot to both of you, really appreciate it!