  Majorization uncertainty relations are generalized for an arbitrary mixed
quantum state $\rho$ of a finite size $N$. In particular, a lower bound for the
sum of two entropies characterizing probability distributions corresponding to
measurements with respect to arbitrary two orthogonal bases is derived in terms
of the spectrum of $\rho$ and the entries of a unitary matrix $U$ relating both
bases. The obtained results can also be formulated for two measurements
performed on a single subsystem of a bipartite system described by a pure
state, and consequently expressed as uncertainty relation for the sum of
conditional entropies.
