Diagonalize operator A by matrix S

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Suppose, i want to diagonalize operator A by matrix S (A'= S^\\dagger A S). Do i need to form S from "NORMALIZED" eigenvectors? I checked and found that even S formed from not-normalized eigenvectors works.
 
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If S isn't normalized, you will get a diagonal matrix. However, the diagonal elements of the matrix will not be the eigenvalues of A.
 

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