  We consider the symmetric tridiagonal matrix-valued process associated with
Gaussian beta ensemble (G$\beta$E) by putting independent Brownian motions and
Bessel processes on the diagonal entries and upper (lower)-diagonal ones,
respectively. Then, we derive the stochastic differential equations that the
eigenvalue processes satisfy, and we show that eigenvalues of their (indexed)
principal minor sub-matrices appear in the stochastic differential equations.
By the Cauchy's interlacing argument for eigenvalues, we can characterize the
sufficient condition that the eigenvalue processes never collide with each
other almost surely, by the dimensions of the Bessel processes.
