Compatible Operators: Same Eigenvalues?

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In summary, compatible operators do not necessarily have the same eigenvalues. While they may share the same eigenvectors or eigenspaces, their eigenvalues can differ. This can even occur when one operator has multiple eigenvalues for the same eigenvalue of the other operator.
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M. next
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should compatible operators have the same eigenvalues??
 
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They have the same eigenvectors (more precise: same eigenspaces), their eigenvalues can be different. A trivial example for any operator A is the same operator with some prefactor: 2A, 3A, ...
 
  • #3
mfb said:
They have the same eigenvectors (more precise: same eigenspaces), their eigenvalues can be different.

No, they don't have to have identical eigenspaces, they have to be diagonalizable in the same basis. If you want to express that in terms of eigensubspaces it gets more complicated, because eigensubspaces can be a proper subspaces of an eigenspace of the other operator only, no identity required.
 
  • #4
Ah, right, one operator can have different eigenvalues for the same eigenvalue of the other operator.
 

FAQ: Compatible Operators: Same Eigenvalues?

What are compatible operators?

Compatible operators are operators that can be applied to the same vector in any order and will result in the same outcome.

What are eigenvalues?

Eigenvalues are the values that represent the scaling factor of an eigenvector when operated on by a particular linear transformation.

How do you determine if two operators have the same eigenvalues?

If two operators have the same eigenvalues, it means that they have the same set of eigenvectors. To determine this, you can find the eigenvalues of each operator and compare them. If they are identical, then the operators have the same eigenvalues.

What is the significance of two operators having the same eigenvalues?

When two operators have the same eigenvalues, it means that they share a common set of eigenvectors. This can be useful in solving systems of linear equations and understanding the behavior of linear transformations.

Can two operators with different eigenvalues be compatible?

No, two operators with different eigenvalues cannot be compatible. This is because they would have different sets of eigenvectors, and thus, applying them in any order would not result in the same outcome.

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