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I read the textbook and found that common eigenfunctions are even possible for degenerate eigenvalues.

Let's say operatorsAandBcommutes and eigenvalueaof operatorAis N-fold degenerate, means that there are N linearly independent eigenfunctions having same eigenvaluea. These eigenfunctions are not necessarily common eigenfunctions of operatorB, unlikely to non-degenerate eigenfunctions that are always common for commuting operators. However, the textbook said that we can always find a set of new N linearly independent eigenfunctions of eigenvalueathat are common for operator B by linear combination of old eigenfunctions ofa. It is really fantastic theorem but I couldn't find its proof.

Could you please give me the proof of this? If I fully understand the proof, then I will be really confident that commuting operators have common eigenfunctions for all their eigenvalues, even some eigenvalues are degenerate.

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# I Common eigenfuctions for degeneracy

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