Quantum Fundamentals 2022
First complete the problem Diagonalization. In that notation:

Find the matrix \(S\) whose columns are \(|\alpha\rangle\) and \(|\beta\rangle\).
Show that \(S^{\dagger}=S^{-1}\) by calculating \(S^{\dagger}\) and multiplying it by \(S\). (Does the order of multiplication matter?)

Calculate \(B=S^{-1} C S\). How is the matrix \(E\) related to \(B\) and \(C\)? The transformation that you have just done is an example of a “change of basis”, sometimes called a “similarity transformation.” When the result of a change of basis is a diagonal matrix, the process is called diagonalization.