Eigenvector-eigenvalue identities and an application to flavor physics

The eigenvector-eigenvalue identities are expanded to include general mixing parameters. Some simple relations are obtained and they reveal an intricate texture of connections between the eigenvalues and the mixing parameters. Permutation symmetry ($S_{3}\times S_{3}$) plays an indispensable role in our analysis. It is the guiding principle for the understanding of our results -- all of them are tensor equations under permutation.