I am newbie for math and linear algebra. Now I am reading about eigenfunction of the Laplacian operator.
If I have $Ax=\lambda x$ ( Let's assume that A is matrix, x is vector, λ is scalar ), I know that x is eigenvector and λ is eigenvalue.
According to wikipedia( https://en.wikipedia.org/wiki/Eigenfunction ), if I have 'Df=λf'( D is linear operator and λ is scalar ), f is eigenfunction and λ is eigenvalue. It is very similar to eigenvector case, so it is okay to me.
However when I tried to search eigenfunction for Laplacian operator, they always say that f is eigenfunction and λ is eigenvalue for below equations! ( The sign is reversed! )
$$Δf=-λf \text { or } Δf+\lambda f=0$$
I really don't know why the sign is changed for Laplacian eigenfunction. I found that above equations are known as Helmholtz equation. So is it just definition that 'eigenfunction of Laplace operator is f and eigenvalue is λ when we have Δf=-λf'? Or is there some derivation that I am missing?
Could you explain about this? Thanks!