  We propose a non-singular representation for a non-Hermitian operator even if
the parameter space contains exceptional points (EPs), at which the operator
cannot be diagonalized and the usual spectral representation ceases to exist.
Our representation has a generalized Jordan block form and is written in terms
of extended pseudo-eigenstates. Our method is free from a divergence in the
spectral representation at EPs, at which multiple eigenvalues and eigenvectors
coalesce and the eigenvectors cannot be normalized. Our representation improves
the accuracy of numerical calculations of physical quantities near EPs. We also
find that our method is applicable to various problems related to EPs in the
parameter space of non-Hermitian operators. We demonstrate the usefulness of
our representation by investigating Boltzmann's collision operator in a
one-dimensional quantum Lorentz gas in the weak coupling approximation.
