In my book they study the following problem :
Let $r \in C^{\infty}([0,1])$, and let $\lambda$ be a real number, then we consider the following DE : $$y'' + (\lambda -r) y = 0$$
We denote $L_{\lambda}$ the vector space of solutions statifying $y(0) = y(1) = 0$, and if $L_{\lambda} \ne 0$ we say that $\lambda$ is an eigenvalue.
I am wondering why they call it an eigenvalue ? I mean for me an eigenvalue is when you have a matrix $A$ and a $\lambda$ such that : $Av = \lambda v$ for some $v \ne 0$. Yet here where is the matrix ?
Thank you !
Generally speaking, an Eigenelement is an element such that a given transformation turns it to a multiple of itself, and the corresponding proportionality coefficient is an Eigenvalue.
In the case of a matrix applied to a vector,
$$Ax=\lambda x$$ associates an Eigenvector and an Eigenvalue. The transformation is described by a matrix.
In the case of your linear ODE,
$$\left(\frac{d^2}{dx^2}+r\right)y=\lambda y$$ associates an Eigenfunction and an Eigenvalue. The transformation is described by a differential operator.