  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.
