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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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    $\begingroup$ 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. $\endgroup$ Oct 29, 2012 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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    $\begingroup$ Why is an eigenvector of finite-dimensional linear operators not a function? Suppose in three dimensional Euclidean space, the operator T has an eigenvector v. v can be thought of as a function that given a coordinate, returns its value at that coordinate. i.e. $v(n) = v_n$ for $n=1...3$. For example, if $v=(4,7,2)$, then $v(1) = 4, v(2) = 7, v(3)=2$. Is $v$ not a perfectly good function? Perhaps I am missing something... $\endgroup$ Jul 18, 2017 at 20:24
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    $\begingroup$ What you write is true. A vector in $\mathbb{R}^n$, for example, can be interpreted as a function on the discrete space $\{1,\ldots,n\}$. There are many good analogies here. For example, the inner product of $u,v\in\mathbb{R}^n$ is $\sum_i u_i v_i$ while the inner product between functions $f$ and $g$ (on $\mathbb{R}$ with unit weight function, say) is $\int_{\mathbb{R}}f(x)g(x)dx$. One can see that the indices play the same role as function arguments. But we don't typically call an object like $v=(4,7,2)$ a function. We call it a vector. $\endgroup$
    – user26872
    Jul 18, 2017 at 22:14
  • $\begingroup$ Thank you for the clarification! $\endgroup$ Jul 18, 2017 at 22:28
  • $\begingroup$ I am glad to help! $\endgroup$
    – user26872
    Jul 18, 2017 at 22:30

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