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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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consider eigenfunctions of the $X$-operator: $X|x\rangle = x|x\rangle$ where $X$ is an operator, $|x\rangle$ is an eigenket of $X$ and $x$ is the corresponding eigenvalue. Then \begin{eqnarray*} \color{red … whence $$(Xf)(x)=\lambda f(x)$$ This is to be expected: $Xf$ is just an eigenvalue multiple of $f$. It seems that the property that $X : f \mapsto xf$ comes from the fact that we used $x$ to denote the …
asked Jun 27 '17 by Fly by Night