Symmetric kernel

Summary

A symmetric kernel is a real-valued kernel with $K(x, s) = K(s, x)$ for all $x$ and $s$. It is the continuous analogue of a symmetric matrix, and the real case of a Hermitian kernel. A nonzero square-integrable symmetric kernel has real characteristic values and an orthonormal system of eigenfunctions, as the Hilbert–Schmidt theory shows. Green's functions of self-adjoint boundary value problems are symmetric kernels.