  We establish an algorithm for a criterion of the diagonalisability of a
matrix over a local field by a unitary matrix. For this sake, we define the
notion of normality of a $p$-adic operator, and give several criteria for the
normality. We study the relation between the normality and the reduction. In
the finite dimensional case, the normality of an operator is equivalent to the
diagonalisability of a matrix by a unitary matrix. Therefore we also study the
relation between the diagonalisability and the reduction. For example, we show
that the diagonalisation of the reduction gives a partition of unity
corresponding to the reduction of the spectrum, which gives a functorial lift
of the eigenspace decomposition of the reduction.
