  Eigenmode analysis is one of the most promising methods of analyzing large
data sets in ongoing and near-future galaxy surveys. In such analyses, a fast
evaluation of the correlation matrix in arbitrary cosmological models is
crucial. The observational effects, including peculiar velocity distortions in
redshift space, light-cone effects, selection effects, and effects of the
complex shape of the survey geometry, should be taken into account in the
analysis. In the framework of the linear theory of gravitational instability,
we provide the methodology to quickly compute the correlation matrix. Our
methods are not restricted to shallow redshift surveys, arbitrarily deep
samples can be dealt with as well. Therefore, our methods are useful in
constraining the geometry of the universe and the dark energy component, as
well as the power spectrum of galaxies, since ongoing and near-future galaxy
surveys probe the universe at intermediate to deep redshifts, z ~ 0.2--5. In
addition to the detailed methods to compute the correlation matrix in
3-dimensional redshift surveys, methods to calculate the matrix in
2-dimensional projected samples are also provided. Prospects of applying our
methods to likelihood estimation of the cosmological parameters are discussed.
