## Why is the cube of a unitary operator = identity matrix?

Hi there,
If A is unitary I understand that it obeys AA+=1 because A-1=A+.

Why does A3=1?
The explanation simply says that "A just permutes the basis vectors"..

It then goes on to say that since A3=1, then eigenvalue a3=1 also, which are 1, ei.2pi.theta/3, and ei.4pi.theta/3. This would make sense to me if I knew why A3=1..

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 What is this operator A we are talking about? Just any random Unitary operator? It doesn't seem to be true for any unitary operator...
 Recognitions: Gold Member You must have a specific operator in mind because that's not true in general.

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## Why is the cube of a unitary operator = identity matrix?

 Why does A3=1? The explanation simply says that "A just permutes the basis vectors"..
Because A takes each of three basis vectors into the next one, so if you apply A three times you'll get back to the original situation. For example, A3 maps ijki.