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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.

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. An operator's eigenvalues provide an invariant characterization of it through its trace, determinant, and characteristic polynomial.

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