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Suppose we have hermitian matrix $H$, matrix $A$, composed of eigenvectors of $H$, such that $\langle A\mathbf i\mid A\mathbf i\rangle=1$, where $\mathbf i$ is the $i$-th column of matrix $H$.

  1. How to prove that $A$ is unitary?
  2. $H=ABA'$ ($A'$ is conjugate transpose matrix, $B$ is diagonal matrix, diagonal elements are eigenvalues of $H$)?
  3. $H^n=AB^nA'$?
  4. $B=A'HA$

Thanks much for any help!

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