  We investigate the process of eigenvalues of a symmetric matrix-valued
process which upper diagonal entries are independent one-dimensional H\"older
continuous Gaussian processes of order gamma in (1/2,1). Using the stochastic
calculus with respect to the Young's integral we show that these eigenvalues do
not collide at any time with probability one. When the matrix process has
entries that are fractional Brownian motions with Hurst parameter H in (1/2,1),
we find a stochastic differential equation in a Malliavin calculus sense for
the eigenvalues of the corresponding matrix fractional Brownian motion. A new
generalized version of the It\^o formula for the multidimensional fractional
Brownian motion is first established.
