Majorization uncertainty relations are generalized for an arbitrary mixed quantum state *ρ* 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 *ρ* 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.