  We consider a system of noncolliding Brownian motions introduced in our
previous paper, in which the noncolliding condition is imposed in a finite time
interval $(0,T]$. This is a temporally inhomogeneous diffusion process whose
transition probability density depends on a value of $T$, and in the limit $T
\to \infty$ it converges to a temporally homogeneous diffusion process called
Dyson's model of Brownian motions. It is known that the distribution of
particle positions in Dyson's model coincides with that of eigenvalues of a
Hermitian matrix-valued process, whose entries are independent Brownian
motions. In the present paper we construct such a Hermitian matrix-valued
process, whose entries are sums of Brownian motions and Brownian bridges given
independently of each other, that its eigenvalues are identically distributed
with the particle positions of our temporally inhomogeneous system of
noncolliding Brownian motions. As a corollary of this identification we derive
the Harish-Chandra formula for an integral over the unitary group.
