Eigenvalues of O: Find Hints Here

Thread starter ee7klt
Start date Nov 1, 2004
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Eigenvalue

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If an operator O satisfies O^4f(x) = f(x), it implies that O^4 acts as the identity operator, meaning all eigenvalues of O^4 are 1. Consequently, the eigenvalues of O must be the fourth roots of unity: 1, -1, i, and -i. If O has an eigenvector f with eigenvalue λ, then O^2 will have eigenvalue λ^2. This means that the eigenvalues of O^2 will be the squares of the eigenvalues of O. Therefore, the eigenvalues of O^2 are 1, 1, -1, and -1.

Nov 1, 2004

ee7klt

hi,
if an operator O has the property that O^{4}f(x)=f(x), what are the eigenvalues of O? any hints on how to go about this?

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Nov 1, 2004

Galileo

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Since O^4 is the identity, it has all eigenvalues one.
If O has an eigenvector f with eigenvalue \lambda, what can you tell about O^2

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