  A system of one-dimensional Brownian motions (BMs) conditioned never to
collide with each other is realized as (i) Dyson's BM model, which is a process
of eigenvalues of hermitian matrix-valued diffusion process in the Gaussian
unitary ensemble (GUE), and as (ii) the $h$-transform of absorbing BM in a Weyl
chamber, where the harmonic function $h$ is the product of differences of
variables (the Vandermonde determinant). The Karlin-McGregor formula gives
determinantal expression to the transition probability density of absorbing BM.
We show from the Karlin-McGregor formula, if the initial state is in the
eigenvalue distribution of GUE, the noncolliding BM is a determinantal process,
in the sense that any multitime correlation function is given by a determinant
specified by a matrix-kernel. By taking appropriate scaling limits, spatially
homogeneous and inhomogeneous infinite determinantal processes are derived. We
note that the determinantal processes related with noncolliding particle
systems have a feature in common such that the matrix-kernels are expressed
using spectral projections of appropriate effective Hamiltonians. On the common
structure of matrix-kernels, continuity of processes in time is proved and
general property of the determinantal processes is discussed.
