  In this paper, using a formula for the minimal type-I seesaw mechanism by
$LDL^{T}$ (or generalized Cholesky) decomposition, conditions of general
$Z_{2}$-invariance for the neutrino mass matrix $m$ is obtained in an arbitrary
basis. The conditions are found to be $(M_{22} a_{i}^{+} - M_{12} b_{i}^{+}) \,
( M_{22} a_{j}^{-} - M_{12} b_{j}^{-}) = - \det M \, b_{i}^{+} \, b_{j}^{-}$
for the $Z_{2}$-symmetric and -antisymmetric part of a Yukawa matrix
$Y_{ij}^{\pm} \equiv (Y \pm T Y )_{ij} /2 \equiv (a_{j}^{\pm}, b_{j}^{\pm})$
and the right-handed neutrino mass matrix $M_{ij}$. In other words, the
symmetric and antisymmetric part of $b_{i}$ must be proportional to those of
the quantity $\tilde a_{i} \equiv a_{i} - {M_{12} \over M_{22}} b_{i}$. They
are equivalent to the condition that $m$ is block diagonalized by eigenvectors
of the generator $T$.
  These results are applied to three $Z_{2}$ symmetries, the $\mu-\tau$
symmetry, the TM$_{1}$ mixing, and the magic symmetry which predicts the
TM$_{2}$ mixing. For the case of TM$_{1,2}$, the symmetry conditions become $
M_{22}^{2} \, \tilde {a}_{1}^{\rm TBM} \tilde a_{2}^{\rm TBM} = - \det M \,
b_{1}^{\rm TBM} b_{2}^{\rm TBM}$ and $ M_{22}^{2} \, \tilde {a}_{1,2}^{\rm TBM}
\tilde a_{3}^{\rm TBM} = - \det M \, b_{1,2}^{\rm TBM} b_{3}^{\rm TBM}$ with
components $\tilde a_{i}^{\rm TBM}$ and $b_{i}^{\rm TBM}$ in the TBM basis
$\mathbf{v}_{1,2,3}$. In particular, for the TM$_{2}$ mixing, the magic
(anti-)symmetric Yukawa matrix with $S_{2} Y = \pm Y$ is phenomenologically
excluded because it predicts $m_{2}=0$ or $m_{1}, m_{3} = 0$. In the case where
Yukawa is not (anti-)symmetric, the mass singular values are displayed without
a root sign.
