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

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 means that the eigenvalues of A3 are 1, ei.2pi.theta/3, and ei.4pi.theta/3. In summary, A is a unitary operator that obeys AA+=1 and A3=1 because it simply permutes the basis vectors. The eigenvalues of A3 are 1, ei.2pi.theta/3, and ei.4pi.theta/3.
  • #1
km707
6
0
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..

Many thanks in advance!
 
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  • #2
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...
 
  • #3
You must have a specific operator in mind because that's not true in general.
 
  • #4
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 i → j → k → i.
 

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

1. What is a unitary operator and why is it important?

A unitary operator is a mathematical concept used in linear algebra to represent transformations that preserve the length and angle of vectors. It is important because it allows us to analyze and manipulate mathematical systems in a more efficient and intuitive way.

2. What is the cube of a unitary operator?

The cube of a unitary operator is the result of applying the operator to itself three times. In other words, it is the composition of the operator with itself three times.

3. Why is the cube of a unitary operator equal to the identity matrix?

This is a fundamental property of unitary operators. Since unitary operators preserve the length and angle of vectors, applying the operator three times results in the same vector as the original. This is equivalent to multiplying by the identity matrix, which leaves the vector unchanged.

4. How is the cube of a unitary operator related to quantum mechanics?

In quantum mechanics, unitary operators are used to represent physical transformations such as rotations and reflections of quantum states. The cube of a unitary operator is important in this context because it helps us understand how these transformations affect the quantum states.

5. Can the cube of a unitary operator ever be different from the identity matrix?

No, the cube of a unitary operator will always be equal to the identity matrix. This is a fundamental property of unitary operators and cannot be changed. Any other result would violate the definition and properties of unitary operators.

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