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

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