Do compatible operators have the same eigenvalues?

M. next · Nov 28, 2012

should compatible operators have the same eigenvalues??

mfb · Nov 28, 2012

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, ...

Jazzdude · Nov 28, 2012

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.

mfb · Nov 28, 2012

Ah, right, one operator can have different eigenvalues for the same eigenvalue of the other operator.

Do compatible operators have the same eigenvalues?

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