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What is the difference between the eigenfunctions and eigenvectors of an operator, for example Laplace-Beltrami operator?

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Real or complex (or vector) valued functions on a space form a vector space. The Laplace-Beltrami operator is a linear operator that acts on this vector space. Its eigenvectors are also called "eigenfunctions" because the "vectors" are functions. –  Jonah Sinick Oct 29 '12 at 4:44
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An eigenfunction is an eigenvector that is also a function. Thus, an eigenfunction is an eigenvector but an eigenvector is not necessarily an eigenfunction. For example, the eigenvectors of differential operators are eigenfunctions but the eigenvectors of finite-dimensional linear operators are not.

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