# Eigenvalue problem of the symmetric real operator which corresponds to the symmetric positive definite matrix

I have a real symmetric function $C(x,y)$ defined on $x,y\in[0,\infty)$, i.e. $C(x,y)=C(y,x)$.

I want to solve the eigenvalues problem, i.e. find eigen values and eigen functions: $$\lambda \psi(x)=\int_0^\infty C(x,y)\psi(y)dy$$

This operator corresponds to the covariance matrix such as $C_{i,j}=Cov[x_i,x_j]$, i.e. it's positive definite, real and symmetric. In the functional case, I'm not sure what is the analog of positive definite. I'm trying to extend the covariance matrix into the infinite dimensional (continuous) case from the matrix version.

I'm looking for references on eigenvalues problem solutions, software packages or papers.