What happens to a kernel matrix if you divide each row by its sum?

Question

Assume we have a kernel function k(x,y), and calculate the kernel matrix $ K = K_{ij} = k(x_i,x_j)$ for a finite dataset consisting of m points. One interest of mine is to calculate the eigenvalues for the integral operator induced by the kernel k. I have stumbled upon the technique to divide each entry of the kernel matrix by its row sum in order to make the matrix independent of the number of datapoints. My question would be: What happens to the eigenvalues and to the matrix itself if I do that? The other technique I know is approximating the eigenvalues by the equation $ \lambda_j\phi_j(x') = \int k(x,x')\phi_j(x)\text{d}\mu(x) \approx \frac{1}{n}\sum_{i=1}^n k(x_i,x')\phi_j(x_i) $, which uses $1/n$ as a scale for the number of datapoints.