  Matrix Szego biorthogonal polynomials for quasi-definite matrices of measures
are studied. For matrices of Holder weights a Riemann-Hilbert problem is
uniquely solved in terms of the matrix Szego polynomials and its Cauchy
transforms. The Riemann-Hilbert problem is given as an appropriate framework
for the discussion of the Szego matrix and the associated Szego recursion
relations for the matrix orthogonal polynomials and its Cauchy transforms.
Pearson type differential systems characterizing the matrix of weights are
studied. These are linear systems of ordinary differential equations which are
required to be monodromy free. Linear ordinary differential equations for the
matrix Szego polynomials and its Cauchy transforms are derived. It is shown how
these Pearson systems lead to nonlinear difference equations for the Verblunsky
matrices and two examples, of Fuchsian and non-Fuchsian type, are considered.
For both cases a new matrix version of the discrete Painlev\'e II equation for
the Verblunsky matrices is found. Reductions of these matrix discrete Painleve
II systems presenting locality are discussed.
