It is a hard and important problem to find the criterion of a set of positive-definite matrices which can be written as reduced density operators of a multipartite global quantum state. This problem is closely related to the study of many-body quantum entanglement, which is one of the focuses of current quantum-information theory. We give several results on the necessary compatibility relations between a set of reduced and global density matrices, including (i) compatibility conditions for the one-party reduced density matrices of any $({N}_{A}\ifmmode\times\else\texttimes\fi{}{N}_{B})$-dimensional bipartite mixed quantum state, (ii) compatibility conditions for the one-party reduced matrices of any $M$-partite quantum states with the dimension ${N}^{\ensuremath{\bigotimes}M}$, and (iii) compatibility conditions for the two-party reduced density matrixes of $(2\ifmmode\times\else\texttimes\fi{}2\ifmmode\times\else\texttimes\fi{}2)$-dimensional tripartite mixed quantum state.