  We consider the noncolliding Brownian motion (BM) with $N$ particles starting
from the eigenvalue distribution of Gaussian unitary ensemble (GUE) of $N
\times N$ Hermitian random matrices with variance $\sigma^2$. We prove that
this process is equivalent with the time shift $t \to t+\sigma^2$ of the
noncolliding BM starting from the configuration in which all $N$ particles are
put at the origin. In order to demonstrate nontriviality of such equivalence
for determinantal processes, we show that, even from its special consequence,
determinantal expressions are derived for the ensemble averages of products of
characteristic polynomials of random matrices in GUE. Another determinantal
process, noncolliding squared Bessel process with index $\nu >-1$, is also
studied in parallel with the noncolliding BM and corresponding results for
characteristic polynomials are given for random matrices in the chiral GUE as
well as in the Gaussian ensembles of class C and class D.
