Could someone check if the solution of the problem is right?


Let $A, B \in \mathbb{C}^{n\times n}$ be selfadjoint ,such that $[A,B] := AB − BA = 0$ Show that there is a unitary matrix $U \in \mathbb{C}^{n\times n}$ such that $U^*$A$U$ and $U^*$$B$$U$ are both diagonal.


Let $D$1 =$U^*$A$U$ and $D$2 =$U^*$$B$$U$ .

As $A$ and $B$ are selfadjoint follows that $D^*$=$(U^*$$A$$U)^*$= $U^*$$A$$U$=$D$1, so $D$1 is hermitian, which also means that $D$1 must be diagonal ?

  • $\begingroup$ No it's wrong. you don't have the same $U$ for $A$ and $B$ in the beginning. $\endgroup$ – Yanko Jan 13 at 15:10

No, it is not right. You started your solution by saying “Let $D_1=U^*AU$”, without saying what $U$ is.

  • $\begingroup$ In the problem is written that U is a unitray matrix. $\endgroup$ – Kai Jan 13 at 15:12
  • 1
    $\begingroup$ @Kai The problem asks you to find such unitary $U$. $\endgroup$ – xbh Jan 13 at 15:16
  • $\begingroup$ @ xbh Ahh. I am very confused about the way of how this problem is defined. Do you think that then I should prove that if U*AU is a diagonal then U is a unitary matrix? $\endgroup$ – Kai Jan 13 at 15:25
  • $\begingroup$ @Kai No. What you are supposed to prove is that there is an unitary matrix $U$ such that both matrices $U^*AU$ and $U^*BU$ are diagonal. $\endgroup$ – José Carlos Santos Jan 13 at 15:32

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