Prove the following:
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Given $r'=Ur$ with $U$ a unitary matrix and $r$ a (column) vector with complex elements, show that the magnitude of $r$ is the same as the magnitude of $r'$.
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The matrix $U$ transforms any column vector $r$ with complex elements into $r'$, leaving the magnitude invariant: $r^\dagger r= r’^\dagger r'$. Show that $U$ is unitary.
Both questions are in my text of mathematical methods for physics (Arfken), but I haven't a solution text.
Best Answer
A. Form r′†r′ = (Ur)†Ur = r†U†Ur = r†r
B. If for all r, r′†r = r†U†Ur, then it must be U†U = 1