A general formula by $LDL^{T}$ decomposition for the type-I seesaw mechanism

By performing an approximate spectral decomposition to the inverse mass matrix of the right-handed neutrinos $M^{-1}$, we obtain a basis-independent formula for the type-I seesaw mechanism. Mathematically, it is based on the generalized Cholesky (or $LDL^{T}$) decomposition of the symmetric matrix $M^{-1} = L D L^{T}$, with a diagonal matrix $D$ and a lower unitriangular matrix $L$. Since the diagonalization of $L$ can be inverted without solving cubic equations, the formula will be useful to investigate general properties of the mechanism, such as flavor symmetries, generalized $CP$ symmetries, and fine-tunings.