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A linear operator (including a matrix) acting on a non-zero *eigenvector* preserves its direction but, in general, scales its magnitude by a scalar quantity *λ* called the *eigenvalue* or characteristic value associated with that eigenvector. Even though it is normally used for linear operators, it may also extend to nonlinear operations, such as Schroeder functional composition, which evoke linear operations.

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This is often confusing to people getting acquainted to QM and you need to stare at it for a while and convince yourself about how it works. Firstly $\sigma^{x,y,z}$ are the Pauli spin matrices and $ …
answered Apr 8 '13 by Siva